Goras, Liviu - Semnale circuite si sisteme

:eTrc8 as FrnsPu HlspecE
nJ 1rdEJ rr! 'euTDTxo uU p1TUTJ 'erecereo TTltml letm e1er6e1u1
'Bfnu e18a EsuT6Txo
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sXee peuT6lro au1{uoc erec lueubes TNPcTro E nes Texe .reel?1ne:6,,
?ucsl?Tuee.pTTAoomalseg.r'eTat leXlsTeucne rtc F:apTuTE16n6eu1FreolTlPuulTnI leTraT6lTePeln6peErIiPbrtJampuunTndosT€afJ,roePcpTlsETleua?TrTJpuuxTBn

ed
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I=(U)d re1nc11:ed u3

'I prnsgu aJE 0 Tnlound aulfuoc e.:ec argflnu €3Tro T€ l=([o])rt'PInu

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'0=1

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t l'l{epls,huag} alqTun pldsarl sTlrunl ep pl?reuab smsgn
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tsr) r3x4 , lxlJ-txl"e;rii '" ', {x)oet"'r (x)zsr 1xlre
:(x)J ad pzeau-txorde
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v

(?t l (v) rl=dpJ

'?uapr^g

tEr) fsl dre [=no"i Itll$t$ Is iltfilllS']lillls
Iilt3ills'illllndt'tntul'lllll0l tr-zT

$rult, etmulTt st slsT[it L2-2, iutllilt tnfin, intctlil, sTtsclml stall
2.
Ittado ft) =r(o) (nl
F1g
utll1z8nd dlstrlbutl"-1,., urr*, (78)
ant
i t trlds ( s) =i-, trl$ (r) dr=r(o)
I
Fomal, se pot scrle relalille frecvent utlllzate in nanewarea ca
I'funclien a 6(t) a lul Dlrac: 1.5 I
dlstrlbutlet
I
W=o ,( s) (tl6s=1 (?el 1lntu
[_o
l
1,5 ilisuri generate de functii scul
oP€rr
o ndsurl genaratE de o functie scari este o nHsuri dlscreti, care
cor.espunde, conforn i.naglnll lntultl.ve prezentate nai eus, incirclrli lLnil
axel reale, in flecare punct, cu r'greutEgi pozitlve sau negatlven egale
clr vaylmoaroebaesearvltauLausl tifnelpcun[ cstuulnraesppoeacttelvtla functlel scarf,. rl&
lntegrald
prtvtttr ca o gi arr
particrrlar[ in raport cu o anrniti uisurE, De exemplu suaa
It ctro
s=fi t (tr) n,
(80) &e€!

L

poate tl priviti ca reprezentAnd lntegrala

s= I_t (c) dr,( d = I-_:t lr) p- mrO (t- t rl dt= (81)
E {-t,8)mro (t-t)or=F !-rttimro te- trtdt=Tt(tint

unde functla n care genereazi nFsura este nuli de la nlnus lnftnlt pAnI
ld t, constantf, Sl egall cu m, de la tr- la t"-, constanti
cu tra+n2 de la t= Ia t"_r... Sl egal6

1. 0ssEBceTxE o misurd oarecare presupune 'i,ncircareatr axel cu o
dfpflieninniatseslptlanuitnnellcapdtetueilnec"cgudtreee"legucrtaedotuenett'adrlntlesaucasionttanfetateilc.nel9nus1icttua8alttae"ndlanisa1cuteoirralrcuaIutirln1pudl.olers"negscmluteinl'ritgl srediuatpt6eogald.ntei

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TTUTT sp PTTqerPunueu elsxTutJu[
cp FITqErnmu axelTurJuT o ni' aJTr?.Eu o-rluTrd lEluezeJder rolrredo
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(sr.) sp (s'l)xl=a--(';s''-)'d I 1sy'dp 1"int=sp (s,1)x 1"1nl= 13 y*

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atTured EehTeTlg-an6geqe'I BJnBItt

alErbalq eelcril 9'l

s,o,e- rf -el +1= (r-) z+[l$+r+0. !
0+z-r1 (r ) rlr+ fr I rlpri*'''l p ) rtr+ (? I awl =te I rrpai
'l
-2 -t€ i

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nc lroder ug [€'O] TnTurlralq Bd 1=(1)] u1p elerbaluT ulTncfuc

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pleluazarda: puns

l{.
+

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?l=

d

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Imtilus'Illfililt'ttltut'ffil'titi s€-zI illlsts ts lllmuJ 'llflIts

$I[IILB, CIACUITB $I $IET[If[ t3-r xnilHTB ![ tltl,t?t t0[ctI0tALA

ETEMENTE DE ANATIZA FUNCTIONALA Pr
Pr
SPATIT NETRTCE TUZITIARE, SPATE HILBERT 3
d
Norma 4
a
Spatii cu produs lntarn 5 .l
spat,it Hifbert 7
5l
AigoEitwui de ortogonaTizare Gran-Schnldt I sl

Norma g;l netrica Lntr-un spaSiu HiLbert 9 s.l
faotrena proiectiei in spatii Hl-Lbert separabiTe
Suna directd de spatti Htibert 11 c(

AFT,TCATTT T/lffRE SPATIT IIETRTCE LIITTARE 12 tI
15 fl
lransfarfrdrt liniare Lntre spatil" normate 15
16 !n
FuneSionaTe linLare contLnua pe spatii netrice . 17
18 fr.
?uncfionale seschillnlare pe spatii Hi].bert 19 4
Aplicalii dlferent iahiTe 21
Iltninizaraa wel functionafe df
))
OPERATORT LIXIARI tr
23
Algabra nomtatd a operatariTar 24 9f
24 r€
Subspagll gl alenente invariante aLe unui oparator Tiniar 25
4
AdJunctul unui olnrator 29 ff
autoadjuncti gi operatori Wzltivi
aparatori izmetrlcl gi unltari &
si-splt gi operatorl nomal"i
Operatari EJ
operatorl
Analagie i,ntre aperatorit Tlniarl gi nunarefe conplexe 1r

Aga flun am artrtat in capitolul ilMulti-nl, relat,ll, apllcatii, ct
gtructurlr', o mulflne, in particuLar o nultime de semals, poate fi
sl
inzeetrati'cu:
a(
elen-enotestlreucntuurl-dtlanligl egbirliscadu(cianretrpeeramcletestdeeaflgnilreealeumneonr toepleeruanteii iinatlrtee
rul$ini 'rpeste care't este def lnittr structura algebrlctr), FE

- o toWLogie, dacl o coleclie de submultl-url ale nultimil initiale u
inpreunl cu operatli speciflce rmltirnllor. Topologla, lndustr de
€
. '' tn legtrtur[ cu terminologia, i.cnumoopderlrgartuiirloes,lnpterirnnestgruicetxutre[rnsee i-E

lntElege nultimea lnitiat5 inpreuntr t.

.qFalzcuul ntouplgoi-lmoagl"llpoersstea"ucmardessuurlnlot rd. esfulnbiqteulatlcneisleteacodrelnapuurnmzft,rtsoaarue, i.n
operalllle
lnzestrattr gl

acr&.r loeeetcruucatucerfs,,telan.teUlneagoinrldinpsrtrlnsestsnprucnteurcft,r o nrultlne a foet
nr.mal operatllle
aou lnlroduse sau submul}Lmlle gl operatltle aduisa crr acestea fu caeul
topologiilotr sau n{surllor.

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sttlllx, cnculil $r 8lsfl$ 13-r tunnl lt tifint nmlr0ilu

sp tiu de eqnala g1 tenrcnul de elanent cu cel de sennaL.
Partlcularlzlrlle vor fi studlate detallat in capltolele congacrate

spa9lilor de s€nnale.

1, $PryU HtrnICB LlililnB, $PAIII HIIEIRT

l. Da pfffu llElnfc lJxltn un spat,iu llniar topoTogtc (L,r) este un
spagtu llnlar L (peate un c6npr de ecalarl, de oblcel C), cu
proprletatea ci pentnr orlce elenente x gi y dln L 91 orice
dln C, (x,y)->x+y gl (a,x)->ax sunt apllcatIl elenent a
contlnue pe IxL reepectiv
p€ OrL in raport cu topologla r deflnite pe L. (Poate ft utlll
rewderea notlunll de continultate. )

Un Epatlu llniar tapoToglc pentru care tognlog,la etta lndueE da
o retrled d se ntmegte EIntiu retric llntar gi se notaazE cu (L,d).

se poate dcostra ci dacf, retrica este lnvariantl, adicl
d(x,y)=d(x+P,y+p) pentru orlce x,y,p€x continuttatea ln origlne acigurE
contlnuitatea Ln orlce punct.

2. Brz; rrn-I,l spAfru HElnIc rJHAR o nulgine 0=1...,01,...) de
eluente dln L est€ o bazf, care genereazd spaglul metrl-c L dacf,
aceeta consttr dln toate gclwdblntnlatntitttleeleTlntulatrweofringtltreurtpToosrlbclalenlnforyrmenattee

crr vectorl dln I precun
ctrs 6e pot construl crr elenente dln 0. Cu alte cuvlnte L este
tnchlderea rultlrll care constt dln toate combinatllle Lblare, flnlte,
poelblle de vectori dln 0.

Un spaglu netrlc linlar va contlne doutr tipurl de elmentel
- el*ente care a6 exprlnd drept cublnatil flnlte de eLenante
llnlar lndependente Oi (care fac parte dlntr-o bazl) dln L;

*=E (11

"ton

- Elemente de forna: t2,

;r=1$* aP,*f a,0,

adlci:

fdtx, ai01) - 0, X! - o (31

ILil

'azpq Toun
eTp 8?uauaTa ap Tnsgunu s6eTalul ttloA TnTnTlEdF P aunTBusmiP uTrd =

'R1TUr,[T f,o?PJado un ap

eTdoJdE as Tjol€redo ap -rTS un aJ€c uT Trmseu E EATlBlTllrPc PareTcerde
u3 TTln ldecuoc 'ro11roxe.:ado Tedua6laAuo3 Pa:TuTlap a?TuJad Troleredo
ap ireTuTT nTleds tm-Jlu! TauEou Ea.reJnpoJ?uT 'reTnsTlJPd utr
JnFTsGp ptlTi; e?ua6ra'A(RuouJJ)ouTnaTpnTFdsPndpsuTT€€3luTaJu?aeulanuon1-€roIdX€-urou6TrasAsupo3afaaXfuFa!
Aqariep .:16 grTpp laTdwoJ s?ss FcPp I{cEItEg nTteds alFawnu
actr.ro EcEp JeTUTT n1{eds un
ea (sTrlau T3ap a?ss arec) ?stuJou
nldeds Tg 'FurJou ap
TnTlpds 'e3Tf,Xau
€3rJ1anr nJ JTJlau nldeds puTuaAap nllcadse: oPFanp?usTa

leuuou reTuTT nTleds un-J?uTp a1uauleTa Fnop E Ta$ua-ralTp €lltilolJ R3TBe

t 'xg{q} F!) 'lr(-x! = (tr 'x) p

"t
* plnlFfe?TdqsasuoTareppl{oee?lu1uaaeur:?aurToalrddsu?Tiglnreo&lrTXTe€'uu,TIlTnotBr.sre6oRunnnpTou'prTP1pBeeeoFrJeJolBppds1uldo9ecp3afnu1?gluJFu:e?ndsduPouuoTupoT\e{l upJ'ot}u
"rrBTpTTsna I

TnTtr€ds uTp aETEunT ap ee(mTfou ezeazllereuab eg 'Euuou alBaunu as

AaA t xp' l.,fl +lxl >l.f+xl
trr) a:3e,?sxA llxl. iel=fixel

,0=x <=> g=l:rl ,6<lxl

:a1trdg1a1rdo:d oceyslles a:ec +U <- T: l' 1 a1{ec11de g

gu.I0H I'l

'g?TuTJuT e{rnTsuaurTp nc E
gapTpTol?Tnuula€TApluJuTeesrqrRloxualPnu}sJTnluEuen^1Eploada'wTrEE,[Jnaledplesrcs1uuoocJ r
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u'R!?spTsIElesrna1u'l€u?asJaesdlueau'TodSruaopc 'l

e?uaureTa a?eJBpTsuoo Tl lod aJEa uI Inrpec em61se gc 1n1deg utrrd r
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YTY[0mr[{ vxtlYtv c[ rufilfiIfi N.ET ltaflIsts I$ ilInilIt '[1yil9$

sHu[[, clrculil $I $Isltt3 13-s nHxm !x iltlln rl,[cll0ttil

1.2 $patii cu prdus intern

Ln a@nanellzraallzfaurnecaglocnoanLcter ppturilnul de unghi dln spa91u1 euclldian se face
lntroducerea de pro&ts intetn
notiunlL
lntr-un Epaglu llnlar.
Produsul lntern este o aPllca9ie seschlllnlari" (.,.>:LxL->c cu

proprletXiile":

<xrY7=(Y,t(j* Yx'Yetr'i zl Yx, r z€Lr \da, beC. (6)
<ax+by, z| -a'
(x, zl +b* (Y, !

dln care rezulti consecintele:

(d,xty)=a'(x,Y7 , 1x, ay)=8(xrYl t (xr 0) =0 (7)

gl

.fr u,*,, ho,r, = * E yfaib,<x1, (8)

tn cazul in care: ((21)) (x, x) >0,
<x,x)=0
=) x=0, (e)

produgul lntern se va nunl produs scalar, iar spatiul respectiv, spatiu
prehilbertian. se nal denonstreazdo ci:
l(x, y> lt.(t, xt (y r yl ( 10)

(inegallfatea lui gchwartz) ceea ce conduce la rezultatul fundirnental
cI produsul scalar genereazE o nornl pe baza formrlei:
lxl='/Grt
{ 11}

lar lnegalitatea lui Schwartz se scrle: ( 12)

l(x, y> l"slxllyl

Echivalentul coslnusulul unghlulul dlntre dol vectorl dln spatlul
euclldlan este dat de expresia (in general complex5):

! o functle de dou! varlablle se nunegte seschlllnlarf, dactr egte
ltnlarl intr-o varlabiltr
gi anttliniartr ln cea]"alttr adlctr:
f (ax,bY)=a*bf ( x'Y)

Agt€riscul reprazlntd conJugarea unul ntntrr conplex.

in pa5rt0lcsu<lxa+rc,yp, exn+trauydlo=(-x<,\,xy)>/+<Cy*,(yy), x) +oCb(tlxn,ey7re* zlgullata(ytu,yl >€,,nullf,ol..

B€

T'reou ap eT4enca n3 r0 rolJeA :TrtEnse ep Tnte)lsTs auttqo os eng .l

TnJPcaTr Te rpTeoE InsnpoJd ATsa9sns pugnT
rolaluelsuoe ro:n?n? alp eTnu TroTeA n:Xuad 1B3ap coT PaAP a?Podnu

0.1
3ter ) o=rOrn
T-it tT::l:*l;
p
Fc rn?de; pcTTdut rorTro?ceA prETuTT Eiuapuadep",

roTTroXeaA IE ue{O Tn?upuTurelep puTsoTo; EoTJTJoA aleod es (ezeq Teun
efeluaueTe TnlT?suoe Ealnd re srec).I-N""'I'O=T '{r0} TrolcgA
ap xas Tnr.m Tp luapuedapul rETu[T Ttr.rattpxgc nnp lo.ltlglglull'l0 '7

'{ealplTun nc e1e6a lrms arec gpdlculrd eleuo6elp ed ap
:o1ac ulfdacxa nc aTnu eTeluausla Bleol pUBAP) olelTtm acTrlpu o elsa
eaoooT.rrlderu3apf,nTopaz:poqfesoTeTlunasunopdouJodc'eTcg!.xa?cpTurl"e?uIpTeTtsmunafooITuTBUoTloTcaedde1(?eT6unT[uJoucT

tr[ppr3uJnnpucqua1uI1l'{r!etmxnrsceeFrcmNpTpsexuepaTulqupclea)1pasalTccseTrTrTlenpuuTnTeazPuenaqr€eeeATJaElolsuzxeaeuAroudaTuromPccdTntprsyEafiru'reeflnraecssPooOrtdmeTpccaFauor

TezEq rolTroqJen EaJEuTureXep FJ tlplpXsuoe '(a?TUTJ ?uns aTauns)

(sr ) lVr. . ., Zt 1n>I rT Jrr,tg-<{0rI0>

:TnTnualsTs sexeATozox uTrd acEI eleod es (aluauoduoc
TueIzpq'erTosTTTirTotxrgcaATapzeexqeuTTTurJoallceapA'pTaEPuoeT1su{cautrnIpl
zN Tp:l,ol \T) acordTrer

ITuTJ ro111{eds Tnzpc
ug acordlcar Tazpq roTTxol3aA peJsuTu.:t€lap allurad a1{e1ar PlsEoJH

('r) ' ' "Z'Tp=e{p'f J{rg=<{0'r0>

{'0} TezEq p gcordlcar ezeq al6eunu
.{,';j','I'=ZT' l=,IrQ'"}0a}reEcae0rreFtolnuEz'Euqteolrp,r?sspnppourdTTn,c{
as nTlpd8 tm-rlu! 'I

{. . pNdl}ru ozqg

'ETTqersumu Eleurouo?ro FzPq o au1{uoc gcep llquedes
alsa uETXroqfTqard nlfeds un Ec BzParlguomtp Teu eS 'Flelduoc olgo
apTcctToa:touannddrslu1aeausdoQ611oee1uaroor6x.or1frTroo?ilc1lo'I\ATu"ugtn0ETz1rE5TqurQoI PzEaurol {""r0""}=0 Pcpq't0
gcue3ppElslaTTxdauoncuF1?€BuT-rAou'oT?=ro;r0euIfi'1[*nTu
o Bzpauf,of ?aeuulfnlcTselTulnoEucTopapapq3flalFq1zelesradeJ?nOs1uo{'pelltldiutsPepu3unag-dr?o'QupTu=pT44IT'xP>1T6UFTcxTBaTplrspoelusarl1uneu1Too56oog1cerop
T.rolJaA ap

11euo6o1ro

(€r ) s_iw'rr_c=-b7]7_rk€rl.lxl

nYr0urx0r t?Iltil ru ilmurl :-gT ttu$t$ Is iltn0llc 'tTvtms

$Hflt[[, cttcutn $t $Isilil3 13-z xilnffit n iltllzA tutcTl0[il,A

{-1 ( 17) SEfi]

E or.0r,0r)=o; k=Or1r...rJV-1 2.

Deterninantul acestul slst€n de ecuatii este de forna: {c,
(00,0r) tez
t^.'1-l[.<O0or,,Q00o>t <0r,0r) (00,0ry-r) ( 18) {6'
(0r,Ou-r) {o'

t" (0rr_r,0r) ($s_r,Ou_r se

[($n-r,00) lui

gl se ntmegte Gramianul slstenului de vectori {9.}. adl
(regIunldaepluenl dCenratanelrln) ciarrrnieaansuelatruelauidedteervneincatonrtui l{u0i r}slsetsetenueluchl.ivalenti,
tn
1.3 $patii Hilbert fre

Se nunegte spaSlu Hllbert un spatiu prehilbertian care este conplet 1,1
conslderat ca epagiu netric (lfuitele tuturor glrurilor Cauchy care pot vec

fl construite cu e1€mentele sale apartln spatiului). Orice subspaliu PlE
al unul spa9lu Hilbert este tot spatiu Hilbert. dLt
Scl
orlce elenent x al unul spatlu Hllbert separabil (care contine o
bazi nrrnirabili) se poate exprima in funct.ie de conponentele unei, baze 0:
0 (nu neaptrrat ortogonaLe eau ortonornate) astfel€

t=X sro, ( le)

uultinea {ar} reprezlntE spectruJ elementului x in raport cu baza
{0r}', lar elenentele a, s€ nunesc coeficiangl Fourier (generalizali)
ai dezvolttrril elenentului x l,n raport cu baza {Qr}.
1. 0fsfnvnTff (1) spaiiile flnit djrnensionale cu produs jrtern sunt

spatli Hilbert cicl baza are un nun6r flnlt de elemente gi nu se

pun problerne de convergentE.

(2) tlnitele de gtmare pot fl luate 0, ro sau -o/ +co in funcfie

de nodul de indexare a elenentelor bazei. Se cunoagte faptul cd

ale l5uSl un1a se face dupi un nunir finlt sau lnfinlt nunlrabil d,r valori
dupl cum spatiul este flnlt dlnenslonal sau inflnlt
dfurensional (dar separabll).

? Indiferent daci baza este ortonornat6 sau nu.

ltn /"*t0 "S<tx'=0>-tS<5x"Q>-€x=cn
T6 'o arTTicerlp ad rolexuauod*" "tq"<*tx''*rQff;-i=3xf=fzinr":*::J:1fi:" :n "*
;Enp/=n==O
Ts .0 ueTtcaJTp. ad : lpxTnzsJ TnTnIolcaA esr€zTleuuou
TeXuauoduoc p
"x lnluauaTe uTp ea:eberlxa
" 1'n y"n=r$
.*=,n'llli3?:"#;"::Hffi: :

:uT g?suoo 'lpTuttcg
'{' " "aO'r0} (TEuTp.r€J l$elace
-ue.rgr ap eTeurnu qns ?nssourtc '1nru11:o619 sp euTlTnu o TIrJ?8Ito3 axeod

'rn6Tsap ',pugAE) rteurouo?:o TJolseA
es nnJ arenuTluoc uS ulglpJp '{"''cx'rx}
euT4{utTepnuuedoapEuTT T:eTuTI TroXcsA
ep {glTuT} repcllred ugl HTTq€rEunu
ep pugcaTd

ill$Tuqt$-mJg aruulTsfl0fioxJ0 ap IltEtrIJ06I[

(zzt TTT

<x,rQ>rS 3=tQ<* "0, 3=t0to 3--* 16 '-tg=<o0'"0>

:EuroJ qns errp:dxa eleod as luau€fa ecl:o
'T0=rig uaae '(aJordTo€roXnE) alEurrouolJo roTazEq InzEc utr '3
'luapuadapuT EuTuJelap as ?u€ulaTa erEaBTI
:?TuT:fuT n€s XIrFJ TJ aleod Tszeq Tc e?uaueTe ap TnJpulnu zeo ?sace uI

(rz) . 4x,rg;rQ t=tga*,r0> f=* ! 4x, rgy=rn
:pcTpE

(oe) Jr=HgrD t= atO r{g;,rD t= <*'"g>

:auTlqo sg 'acordToal Tezpq TE luaus1a arecaTJ no x TnT

TTJelToAzep Te Tr$[eu roTTqup Tp rBTpoE lnsnpo:d ATEe3cns pugnT aceJ as

'1t6tr ezeq nc lrodpr uI x TnTnxu€ulaTa EarelToAzep uI uTAra?uT areJ {rD}

l(eszxeeqTdpuaoos:d1Ter:eeur ae6zeuqI)puroglluearazuenrdua'er(t{RI")c0T}pT€aTTl'B1snnlu1Jeneurl1aacplaaTdJnssToPnuuanerc?esuaTTssuEuEJeaelJaoEp'Aq'T{ot0za}r
1unT1ldn:1'aerapdcssr€eeorasulTuuelprauleadpap'auTl srueo1Tus1ual urTopfazapTqlrtT;nJzopIpTuTiIB'desceT;nl zaeloeoudIe'sE {rD}

{rD} rnmul;}Ids wtrruHw&s0 'z

6zpq eap'ZTuTTnBTXauTl raolu^a[arasToJoRfadlnppu'mTesmpTltxuoep?ouut Iu"I"'8Fz-eex'gapu'ZT -a'BZa'Ir-Ec
a'Isnd'Oy:;ar?poudopZroTnp€f .ar€ToelgluuarrunaTuelrdT3fpcTenJTJoetTueozaTTaltcmnsc a{uapuodsaroc u!

z Tg B aTTwFiTnu

rnYmIICt0t Y?t1Yff utfiflIfl '_ET urusls I$ ilIfl3llt 'xlvlm$

$HiltB, Cttc0lil $t sI$TBil[ 13-n BtlHBmt t[ ttilIz,t tuicll0mil SH

* extrager€a dln elementul xk a componentelor pe dlrectlile tn
S",., .0r-, gl normalizarea rezultatulul; 1.
o:
ic-1 (0i,x*l{r $r,=urllu*! (?3)
HJ
u*=xr-E
A!
Se obtlne astfel baza ortonomate {Oa}.

u2 x2

+2

u1=

Fig, l",1- Reprezentarea graficX a principiului
ortogonal izdr ii Grau-Schnidt .

0

1.5 l{orna gi netrica intr-un spatiu Hilbert T

Proprietatea fundanentalr a spatiiLor Hilbert este uruitoarea: C
ftrtr-un apal,tu ltirbert produsul scalar genereazr
pe baza fornrlelor: norra catre, la rtndul I
el, genereazH retrlca,
e
fxf=fixfiF, d (x, y) =lx-yl=,/A7lx-fr l21l

orice spatiu prehilbertlan incompLet poate fi completat pentru a
adHugarea
deveni spatiu Hllberts, prln tuturon rlnitelor glrurilor
aefucpneadstaituneelnlliltlanbrieeterdt.l,npALrosgdtlfu,esblurlsnspecaiantrlleaurlleaLsr,-edl(eecmfoinmelnrpetelaelotarprtauoddl rususgpeaalotelriudslecuafilnlaIinrreidqdd-eisnvetlnr:ee
rt

csnnnapuueuantc9nrihlltoucIycLd).oolSf$malecrepetpgrlceotadteaarietstueutnla{dilsncnepgcmdaeoeotlmietnuesprgrtuulreiaiints)cua.aratmErspexitttlfnosterEataelrnleia.sunafncootaresmtiopzsrtoarecnlmsigue-Si,tr.m:rluiaFeerrllte(rstcircaa(euSnsccsto,ehdfnoy)pdrrieuinenriraLrsr€rsi(,ispnica-lias,ll1ar*eu.li

oF nTgerErnu e4elTuTJuT o llt 'TmTsuertp
nlfude un ET t?EzTTBnue6 ero6r116
TnT Tef,ero8l TnluoTP/tTqoe also
eTirler nlEeoce to [nJrEreU
6

Txrurolsu'rl 1'qep""';ff;m:";"t qffi;:"l:*t1il[ffit"i "*l#
ppetpo:d BeJeArasuoo l?u;zeJder (sz) eTlETer lc rglpXsuoc '{q !
uI :pTRos snpo.rd Po Etts T€
Ie rolalueuoduoc fnTtPds qtardreluT lceq

{szl 4rf 'rfy-fu t ,* trgy=IP I'l <,{'x>

'{g.re f=

:a1sa 'leaesrud TnT ergroel ?zearFTtreud e.rec 'lslfnzer lTP un
o(pnrcLrolruogdzrureurlfsg'ndBeBrppJr)pPerdFSrFtWm srEurotsuPrl o a?se
'flBtrrouollo pzBq
r TnT gerE6.rolsusr?
IITqJnuT1lztnBa?dJdsdEaIrreITpTtellrtErr€6oreruef de"oXTTpsnAoBeIlEeEcrlEufndanzJJTqeonEBrIlrTptnuc?9sTJonorJperBlrlpyeexlTlTEunnuaxogTqctlXsu'eT'olrs€9deJcroJreouelTucTgTtTttuxne!E-lFcTAElcTrqTsnctzetlaooJcc
seuttqo 'c<-:1 puja r<-lx rulued Tauuou TTiflTnufluoc nlTroltp 'Td

,lr"l 3u=vtvtsrplp X 3= <c+,r+>F*I" 3ou3=

= <rfe S>= <ox'ox)=zl'x!
$'*rt&al

;rore $r"x ep,'n "0i'" 3;ii."xrff1=x

uaAt

{ee} ,l <t ''g> I Y=I*t
6lnctTnatteTcdnez::*T6oerp*1-A{aree1lu1s,eetrgplrrdoa(delTTprTgnJsqTTTpElrpEFEsGronHTturao)oxcetilaulEreeatuurlTEospute"oTe}UnoIoTauxqnFoTzfTTEircHquuosneul1eot1dl'1IeT{aIqefsdn{rs'E"$dp'euq'1TBUa0oUT":Aed'q&'I}T='H0I

'(n na e:enuTluot uI

?e1CIu TJ €lt sreo) J uT eTlTu;lep u1:d suep s?ae I Ec luapT^e e?s[

{sz} (sJlrou-tt11*i' (Ex)*ttu' 11.x' o-!? 1.['xv

4ur('ux;t!11o

YfitoltsxM YrIlYrf xl lillml ot-gr ffiilst$ Is iltnill3 'lltil$

$IIII[8, CIICUI1B $I SI$IBTB 13-rr nmxnt D[ t;tl,nt rrucll0nil 5S

1.6 Teorera proiectiei in spafii Hilbert $epar$ile :a

Dezvoltarea unul elenent dintr-un *patiu Hllbert sepa.rabll Ln (.t

forma: 1t

t*E un** lzel o
I
poat€ ft trunchlatd obllntndu-se elementul x1n) de forna: {30}
I
xror =E tr&*'
:
car€ este proleetla lul x pe subspa!,lul Ho generat de baza ortonormatf,
by., .k=L,2,...n, vom lnterpreta el€nentul x{n} ca fllnd o aproxlnare cu I

li nr.ufu nai nLc de dlnenslunl a elanentultii' x. Dorir s{ arf,ttrn ctr I x-
*{ul! este nlntrnd adlctr aproxinarea lul x cu un elenent dln aubspalluI
dchieacsr otd€nesopipdrtoelnlref&,c'uSalaLltneeclolamezeunlntuitnlxuncl adxrlenpeallscoueFbasslptfca,a?allpcurulol\xfilrm(dtaeirosertaaemnsatteapcdrooinlnetscrtetil.etxui.l)tgtr"i

x0:

lx-:ro F -! (x-xr'l i * {xo -xtrl ) !'= {3r}
=( (x-x,o,) -(xo-trr,l), (x*x,n,) - (xe-x1oy) )=

(x-x1o1, J(-X(n) ) -(x-xtr,), xo-x,o, ) *
-(xe-x(o) , x*x(a) ) +(.t(o-x(u) , xo-x1n) >

Observ6n e& termenil dln nljloc eunt null deoarece (x-xlnl )apar9lne
coplmentarei iul llo,lar (xO-x{n}) sacpaalargrelnedllnutrla\eleeoal nctue iacpealretlndAonudl
subspatil sunt
ortogonale, produsele

unor subepa?1i ortogonale sunt nule. Rezult& ctr

[x-x, fa -fx-x,o, 12 + lxo - x cotf" {32)

gi se vede c& i x-x/ este nlnlntr dacl x0=xlnl. ln concluzle, cea nal
bun! aproxlnare a unul elenent dlntr-un spaglu HlLbert cu un elenent

dintr-un subapagiu al acestuia este prolecfla lul x pE subspallul
rgspectlv" Teorena lul Parseval conduce Ia rezultatul:

l0 eu alts cuvinte, nu axistE o aproxisarc nal buntr lntr-un

subspatlu cu nrear mal mic de dLlnanEiunJ" pentru un elenent dat dectt
prolectla sa pe subsBa9lul rasp€cth'. Inverg, o aproxtnare dln ce in
oe ra1 buni a unut elenent se obtlnn adtrugAnd cuponente pe gr$epattl

suplbentare, ff,rS a le nodlfi.ca pe cela detarulnate 1n191a1.

:utrd o?€eutJep ss lp=g p1{tds ug r€Tecs TnEnpord

3I=r

tr€) )'{${xl

ETiTpuoc nc

(e€) .tfft6- =ff 't?S{x 'Htx t ( ",,' 'laxd"eTT'Telxce'rTx1ox1d7; "lg
:(Ig
a11lfeds

uTp €?uermlle op aleuoPro aluaAaee puTTJ H TnfnTleds ofelugrfirTe 'he+t ,1

'h UEqTTu ro111feds e FXcerTp Prns Bc H l:eqTTH TnTtPds uftrtlec

'TnTnnuTluoc rerelnd eP 'T1{lPuoc

elTnrue u! 'TJ ealnd pA UaqTTH TTicds ap TTTermrgtplrutB. Tunr1TPleUdTcPJpPcO 'n1zrg1
nTo"ertelrupe'Tdrges't?pmerdseTeplTTqqe'€ueorltcpupugynlr€etxEltrtoeTdlHTutfttJleppTduWsnrt{Tuhnlps} lndecug
r:aqr1g TT{uda ap cut4Tnu
osJE! TEs' UeqTTH nTlpds
rm Trulsuoc relnd mrt :EsJeAuT euelqord exmuT?uoc u1 uetmd eg

'eTB8 erE (aTEuo6olxo) TTtPdsqns ap ExcsrTp Pmls

pufTl Eurn urp B18se 'lup UaqfTH nT4ede tm-rXqu UaqTTH TT{pdeqns
EtruepTAs rg etmd E op lEzTn
s€rlE?TTTqTsod e Bns TElr sp €T{nc8Tq

uefiIII TTluds op PlcarTp um$ ['l

"*til-" 'ta)ax trltxl I={

3-rFi
t5€) Jgetr-g 'gag tvte {rlgl+rlel'.lxl
: $BrobElTd TnT Eetroelil eJBJ:sT?es
{rg=x nT?cadse: sil TS tee n3 qp+EBElcxolruTepupomlan6aacxTsfa,oUXT€cctumI lTpet(r[l*s:p{

ft{rdgqns g aTlcedeer Enop roTec
n:ruea g=4[ailyr 'halx nps 'geq Tf tseE A) e1euo6o1:o TTtedsqns
o=<q'p> es usqITH n1{ede un-rlrg lJPq
slTnu TEE nss Enop P{uepTna u1 tmd

ttt=rl(')x-xl
(rcl ,lrrl

no r ,{[is Tn?usuoTa Tg x rnluausTs eJlurp 1e{ua:e31p P&rou urrd (u}x
ed pt4ulxordu lntopJ o orBs sd TTIoxa coleTceldB allnrad ou ef,E3

(s€) ,lrel f'rJ F)xtr

vfiIol$mc Y?IlVlt xl lltlEnl 'T-gT ffiugts I$ ilIflcilo 'nvffis

$mil[[, cltcutfi st Etst[tB l-3-t. rrn{Br?[ Ix ilAr,luA rurcrl0mtA

(x, y2 u=fi(xy, yx) a, (38)
91 lnduce in ctod natural norna

,*,"=,lEo-*^= p (x*,xr)r, t3e)

respectlv netrLca

pd (x, yl x= d, (x*, ypl a,= E ,r*-"*, xf yr) n* ({o)

Pentru un k fixat, Ln cazul spatil.Ior Hllbert tL separabile, un
elment xr.etL scrie, in functie vectoril
se de unei baze ortogonale din
lt in forma:

**=E 0*(0r*, xr) {lr)

tn ,particuJ"ar, elenentele spatillor t1 pot fl funct.li de una sau nal

nulte variablle"

l.Oasmvafru se poate vaoprbroi ledcetisilrourrdpedlsreubcstEpastlii Ln cazul

prolactllLor oblice adicd avAnd baze

neortcgonale. Astfel, suna dlrect6 a subspatillor H. gi H= conline
toate elementele x=x1+x2, xagHl gl xr€H= unde xr=pr1 este prolectia
(ob1icE) a 1ui x de-a lungul lui tt pe ll'iar xz=P,,r este proleclia
{oblic6) a lui x de-a lungul lul H, pe subepallul H=. Desigrur, i,n

cazu.tr" prel,ectiilor oblice nu mal este reapectatd "teorema lul
Pltagora', tn forna (35).

2. Exe{Pl,{, Consider6n un clrcult electric compus dintr-un nundr

finlt de eleuente uniport lnterconectate i.ntre ele. Este cunoecut
faptul cd energia" dlsLpatd pe flecare ranurtr a circultului se
scrie ca produs scalar sub formi de integrald:

11 snergla (in sens flzlc) consulatf, sau debitattr intr-un lntewal
de tlnp tixat T este lntegrala puterl{ lnstantanee pe acel lnterval de
tl$p. tn cazul unei ramuri dintr-un clrcuit puterea lnstantanee p(t)
de ramurd este produsul dintre tenslunea la borne gl curent:
coegunat* curentului flj.nd aceleagl.
iik(t} sensurlLe lenglunll 9l
I'*{t)=u*{t

€*,HS e3 Ei;

'hq{etb€c8lt?* i"td.Q9d
NA @F{ earl gHfi Hr EssHEEEE8o8it'5 ge6Skg *E e3 i
-o
3gFEI -{=t) *EF{
lll dnF^O+{aU,t{

fle igt ^ tuu Fifi -a€ i
- 3E
E- lil i fif ag*i i iii I i
iEF

-n
()

8.q
i= HE
f!6 gd
T '€tm\t ;
Ihvk frH
t .a
+OJ-i;
ir-l G 'N'd ltt
;r <ri ti; I f's,
$a t$E [;gg$ IEi I ],ob1..l

l.r tr
lO i{
+oooJ fl
'.,, 3EE .'rdI{OEo iilf{{iiiIlEliiigOor{€-iOji''65E,h3
ego0 a++t!J{l,
-fts
o.
^+J.OJ4 t{
€@
E? Id F{
i{ o

a4 1=co55at56{ett03+{ot,rr{ n€C aH8ill*'ldt(SdH'5?$ H": Bu&) t+iokttt

E€ +Hl'rr o "H.iF{+r'{ cro0

<)

at
A

--E o trtt*{ E t{ B

:tr:D,T-€R,__.. -

<A rrofi Frl. (E'r igEE E?:;'€tCO5ad vCcf0 r0!\llldlU.d.oYartirt{€(DOg+OuO,rFO|(rD|tAo o
+Er
=-oe $ o" vUl In +J

N \ ota

*J x U
r-
r <G'c."f *8.''EH Lo;
E
il $;Er{ (l) E8 E.i F'*S;E;:g{ r.!{rt "
-E ts .lJ :$-i i'fi;a
tSd F{O
F (l rGoLl.' to1O6(aogf.{r
H'$: F{ r{
h { !q1F{

1; u, (aEoU'F?+lrit
H- f;fi[, ;fi!.i.g l:gg g N do.at
"< itlooall{ P(lrt x dD r0
tFr:{l
r..) a 3o. E F(-E)Iii: i=tiiEH-l <LD{ T Lv< 'g,5
-fto- t6{
-d FG.{ -JC €5
I cEi-H Toan
.{"sF{ It{ b.r
EF{'.r t.{ iri
tr.9-i Atu.:t |d
sa "t3 X <{-EFJ !l q{ 9c

I (1, e 6;
(d ts
5ul o rr, LG.' 'Fl etd)l*fi;
<L Fl €liiiiEiIiI€EEG&A{ ll
vfrFs{ltnad
-sm toE1.t3d{61-gotJn F
b.{ s6if)r

t{ +--{ trL(trr{}rt .oEo

3o(tl} ;l? rtl ;gcfriFHHi IGo

E +too({l,'l5qtor r.d a Ei-gE EfiiEIlE}-.lfeltrFl d
g
GI t4t
d
.Etr.{t::tt
re{g lii t{ fQir{d+laq!l QO+OolU) (O)iUlC;d.F'til5! d 6€5
(gdGg (\it
+ttQt{,llHEr@{6a+qoElrr.++CrollIOrt}
E-n

$Hil[[, CIICUIn $t $tst[ltt 13-ro [nnltx r[ fi$,I81 nriltl0iltt

Deoarece

lnrlsl4lxl (521

rezulttr ctr norma transform,trril T este Lunginea naxin{ a Iu1 Tx cAnd x

acoperl sfera unitate dln A. o transfornare este n[rginltl daci I f{ <*.

2.2 ltnctionale liniare continue pe spatii netrice
o functlonale thiarl contlnui pe un epagiu llnlar nomat L Bste
o transformare llnlarE partlculard f:l
colnclda cu nuJ.tinea nunerelor complexe) care este:
(5sl
- liniari
f {atr+byl -a.F(x) +bF(y) , Vx,yEL, Va,bec

- contlnu6 3r(- | lr(x) lsr.lxl, Vxe.r, (5r)

- are noraa datl de (551

l^F1l=xgl-trr.plr(x) l=inftr, lr(x) l<rlxll

1. &ssRvrrtr (1) Norma este o funcllonali Ltnlar[ gl contlnui cr!

vaatal(og2ra)l tvinemleRm*f.aa)cnetuldueLosxebjd.reinlnLtreginondoulmulantrryaXnrsutluoirncdmrplileJxlnlFar(rxa)
(funcgionalel) F (norna reguLll prln care se ataseazd nrnere
conpLexe elementelor xOL) adlct naxintrl uoduLuLul lul f(r) pentru

x apar!.lnind sferei unitate.

2. Tnonnn urr lrrsz tn cazul particular ln care L'este un spallu
Hilbert are loc un rezultat extrm de lmportant (teorera lul
Rleez);

?orna generald a unei functlonale liniara gi continue pe un dpaflu
ElTbert aate:
Fs(xl -(g, x> , gew.
(s6)

esadclaoapelCtan&urtnadallneluttenrxeecdlugienvnlSelntntlget e,ifllneoxunrantectenergteelfeludenioncLmSuHpllol,enHrraee.lgidAullcnllnae- laCaderetfEallcnioncdreeoeednapatnoltnnn5dnueddlnestcXpeIrodoifblninr$sttlrrnuEeel

elementele spatlului Hlrbert H gl elem€ntere spallul.ul gdu drraL exlEt[

un tzcnorf,lsn, fieclrel funcllonaLe dln spatlul dual coreapunz6ndtr:-1

un elenent dln H.

$Hillu, cncutTt $t $tsiln 13-tz ttxutn t[ ililIzt flilcll0mr,l

Spatiul dual al unul spatiu Hilbert poate fi organlzat ca spatlu
liniar normat dacX se definegte norna funcflonalei Ps ca fllnd norma
1ui g;

lrel_l9l (sz1

Aceastd definltie este naturali cici se poate scrle: (sa1
lf' (x) l=l(9,*> l<lgl.l*l

e9s1t,edclnhiainr engoarnllataltueal lui Schwartz, rezultX ci valoarea constantel K

g.
1n partlcular, dactr spaiiul Hllbert este separabll, forma generalS
a unei func$ionale llnlare eontlnue pe H este:

r'(x) =E a,r (ss)
n.1

u4de x- 6unt conponentele lui x ln raport ctr o anumitE bazi din tl
(care, dln separabilitatea lul H, rezulti ci este nundrabilS).

2,3 functionale seschiliniare pe spalii Hilbert
Produsul scaLar este o funcllonali gesctrli.iniari (blliniari in
cazul epa?llLor Hllbert reale gdlefyinidtelnpHesti€nRv) aadlolcrtri o apllcatie care
duce nunerice f(x,y)
pereehi de vectorl x
eatisftrcdnd relatille :
f (arx.+arx, ,f) =aif gyy) +a;t(x2,y)
(oo1

f (x, bryr+b^yzl =brt (x, yrJ +b"f (x, yr\ (611

Prin urnare, o functlonalE seschilinlari este liniari lntr-unul
dintre arqunente gl antillnlari in celilalt. o functionali
seschillniari {blliniari) este mirginltE daci existi un num6r raal R
astfel lncXt:
It (x, yl l<{xl . M
162l

Cea nai nare lfurittr inferioartr pentru K reprezintd o normtr pentru
functionali:

YJ;t ff{{:,)r f, ( iifl J :tIf' ryr' = (631

Orice operator deflnlt pe un spatlu Hilbert penrlte dEflnirea unei

functtonale seschillnlare dupd fonrula:
f yl =(As<, y) (er)

^(x,
o functiond.d pdtraticE este o functlonall gegchillniard

prpTtr.rl olse Q+xe=(x)v eTtEcTTdv' (tx)v+('x)u*1ex+rx)'Qt ErqceRrcsBprETTPuuTnur
alsa nu aTlsoTTde ep TeIlsp o 'roToluBrdE rEr?uoc 'sTnueu alue?suoc q

rg E nc q+xe=(x)V puuo] ap aTlpJTTdE o a?sa puTlp aTtpJTTde O fi

eT'qaTrlmTeTedTtsoluTeux€T?Tlue€pfcJn€opeTurTBoTzsITE'TpTpTrTTegduT€atoTfnumTzJpecdpuurpya?I rue'ezItTueleeTl.ptlre^auTrsTr.olpc'eeoJp€pJUedT-nrTEpJoFelsIeIlTpAtTTrT.rruaoTplcIsaaApp

{6e) 0= U 0-q ' $l'x) r+q (x) v= {x) J- {q*x) g

iqlxp;tuTT
:puLroJ ap 11:aunduocsep pa1slTTTqTsod 'q

(8e) --mJ- rif:tt ,t0rr'{r
:lelserdxa e alTuTJ e?T[tI Taun EtruBlsTxa 'p
:Trlcun; Taun
uTe! xpeAnTTraplopds."r(eeesslenTppouTro?ruueTTJTonFdrT?TguqagSdTaZTleaToAtTduaeTpAJeTep?ToFsendeosrpafr,aoBrTeTrTlpmToTblPTuudnJefnT/IepnTrzlapeuroa6pu6TtrsTuEoUc)
e?€JapT suoc TaTf pcT ldp pTeTluaralTp ElTunu, rEuT Je €T!€rqxo.rde o-JluT Jd
plcTunedraTlanurna€s €a?BlpuTceA o-Jluf pTe?sacp TTrTmoTuT Ea?plTTTqTsod

TTIETTTdE Tetm aa?ptTTTqpTluaJeJTp 'n1d1cu1rd u3
'orTgeTluaretTp

roTTTleJTIdp aIp (lBp lcrmd Tnun rolTtplprFceA alpTcose)
aTproT :o11{e1al.rdord TnTpnXs lraTqo ldarp are plJueraSlp TnTnrT€C

aIIqPTluaraITp IIlucIIft I'U

'TTETUTI rolT.ro?pJado lnlJeds nc rTrlsuozT Tcap

{t e) lvl-l"tl jl
:guuou
f
nc lpuJou rpTuTT nTlEds un pzpauuol aoTlprlRd a1a1euo1tr3unJ 'IpxnXpu
I
potr ug '?Burou xpTuTT n1{eds un nsnuoJ TJeTUTT g1:olerado unp

{ee} qs<y'xy= (x)tJ

usadT! sT'lleJfTeuuqoTTTlTeiJHpunTnJT1x{'oe1ldnesJ1entdm1o{enndJ:s1eucoeoTdAaT'rlnu.meoTurqoeplealu:ean1puruToTdgscaaoft^aoTlcpuu:nuHoT!Tqlrl:ec[rpqumfTnusToJpIe{guco1tIen1dqTeslrep1rdgosdc

{se} <x ':61'>= (x'x)tJ er€3 nrlued

:I=x (FJeTuTrTq)

ntmllrril wrrtfi It xtxfixlt 8I-EI Itxlst$ I$ [tImIIc 'lfiIt{ts

strHln, cIlc0ITx $I st$iln 13-', lr,iltltTl rx tlil,ut nilctr0tail slHflAr[,

conduce la notlunea de derivati tare. Astfel, daci li-urita 2.5 H
(x+e4L-A (x)
,rtoro, A ,' = ?AAhr'^-r' 1.E
a (701
vi
rt0
pentn
existl, ea reprezinti derlvata aplicallei A dupE direclia h in punctul
x. Daci pentru orlce h€!, existd derlvata direclia h in punctul resPe(
x, apllcatia: dup[
gau, €
h - H(xi =A ix, h) (zrl
deoart
ge va nrnl prina varlalle in sensul Lagrange a 1ul A in punctul x daci, acestt
in plus, A(x,h)=-A(x,-h) . consit

Dac{ e a&nite in punctul x o primd varlalie gi, i.n acelagi tinp, dlrec:
existi un operator llnlar gi n;rginit T*:X, -) X, astfel incAt
U
$ t*l =t*o {,721
deriv,
atuncl T. se nutegte derlvata Gdteaux a lul A in punctul x gi are loc
egalltatea: t
A(x+th) =A(x) +tTx!1+u(x,h) , lims (x,,11) =0,
(73) de fo

Daci, ln veclnitatea punctulul x, apllcalia A se poate scri_e eub D

forna: deriv

A{x+h) =A(x) +r*(A(x) )h+c (x,l:) , ft-o =) c (x,})-0, (711 1

unde f*:t, -) f," este un operator llniar 9i mirginit, acesta se expri.
numegte derivata Fr6ch€t a apllcatlei A i.n x. DgelriveastteelecoGn6ttlenauuXx si func!
coincid daci derlvata Gateaux existi in
Frech6t v
punctul congiderat.
r:nde
Psntru o funcS,le f-:fX(*x-,>+tnh,', teR Si h€R":
xz+t4, . . . este
f (x+9fi1 , +xa+tha) ( 751 lndel

df (x,n7 =fit6+th) lt.o= (ze I t
egal
=fir*, hr* . . ..#(x) ho- (Vof {x} , })

adlcE dlferentiala lul f este produsul scalar intre Eradlentul lul f
gi ht=.

aB vf:=( #o\,...S1oxo

rsTer,oTlarmJ Tn?uaTpPJF na leucgJ:odo.:d q ru?uad 3ct pti€5€ wa1eqtr1e6a \
*
{esi :au'g$qer lsefl=[rylbalsl<ry'sa]l=l (r]s{sl 't{ 8l ?uapuadspuT

as u?Jer.ulos TnT €a?€XTT€EauT {I30
slsa JPp 'g ap apu?dap a?e+d F.lea TaTetioTgtlun] lnqualpe:b e?sa

tTB) qx1 gr{Sg={lr *ff r* *=-ta

:Inf,s?3an apun

qJA'r/> *{ir{{$'rf > ** {*) ;r'0gu*S'r{n *

{os}

= (rr)J€b:rs *= (*)*oo

ru*1e'q as;-To nJ:tru*d x ilT gnuTauop alsa (x);qg Fsep 'a;PnuTluo* rlg

{6r} .r=}o ft - t=lvl ,''4.o ft=o
Euuo3 uT t{ rrTp R?Eul:ouo?io s?sq n3 q:ode: q gmp:dxa as q, Trtro?oaA
'galeuolJoun;

7nlu*rpesfr e1$eunu as JeTeJc snpo.rd €3 PTPuoTlcaJTp PlenTJap pu1:dxa

es fr:sa Rr 1;odpJ uT Inso?rerr '€fPa-r JoTaTBuoTfounJ fnze: uI
x ap p?uapuadapul a?ss p?eAT;ap

'g1ea: pTTq€TreA ap :c111f*un3 TnTnuF3 baleue 'gn un:asqo eg

(aa) r*lell I 4rg'51= <xry5_ eci.+x"r6l u0i-Crl* (xlE(o

:e?sa q eg{aartrp gdnp e1en1-rep '<x'6> Eutf,o; ap

?rrpJ?ppTJnp$TpBraupuusn€d'uEsnaa.a:oTe*T$aTpt1aa'e1u"u:oTnlp$* oaa?uJsr€}o31Taanl{;Bs?nTuuaTTd€Too.xroxula1uleraauJloeAluXdoaeTualnspJeJlfTonpuznETdJa?uuEpInTrap

(rr) {ci} i o'r uT {T= lq l) q PTltarTp
ru;1* {x) ;rry61
rpTnunoJ uT:d WFlEfA.xf a111{eraplsuoo
pdnp elenlrap e x lnlcund
a1euo1{oun} Term
ulTurlap 'totrla1u€ lnSer6eled uI 31n3gl
apuol{run; e"g puezTJpTnsT?f,pd'Ta}lsg "atf,cal1p o gdnp Bro?sece
larde1:ren eualqo:d aund aXeod as 'TJo?3aA ap puldap eTa sJa:eoap
':o1a1euo1t3u'nJ Tnues uI "aunTxaTJuT ap lrund lnun leunu '1en1uaAa 'nes
E.ruTff n€f uTx€$ lnun pundraroc aTa gJBp TfTqe?s as E nrluad aATlsedsaJ
€T€lound uS pTqsffiJ pu€uTl$Exa TB Pz€aTnu€ as PlPATxap arpc n:luad
sTTioTEA prryTle a3e3 as Jotaul*Jlxa €arpuTwa?ap 'p1ea"r FTTqeT-rEA
g:nbuls o ep oTesJ :o111{cuny Tnz€a u; Trutrtrotru$ rg sTsrguJ.ir$ " I

ilTeirCITl$un] Taun eafrBzTHTull{ g'e

flYtsI$[BC YtITYfiy A[ ru$Hgt[ 0;-f;T cgxt$is I$ cilfic&t'J '['trv{H[$

slllll,B, cllculil $t sl$fiil 1-3-r' xrxilnt ![ ttilrut tu[ctl0ttrA $llllllB,

f cind dErlvata dlrectlonall are o vaLoare naxlmE egali cu norna 3. 0l
vectorulul gradlent. Aceasttr observatie este utillzattr in tehnlcile
o
nrrnerLce de deternlnare a extremelor functionalelor pe baza evaluirll
Ilnla.:
valorllor funcSionalelor in dlrectia vectorulul gradient i,n puncte T:A'>j
consecutlve. Este utlllzati ideea cd cel nai rapid nod de a atinge un
extren este de a face ttpagltt in direclia vectorului gradient care Tsi

corespunde variatlel uaxi-ne a funcglonalel. folos

2. FOmrS PAIRATTCE tn cazul funcllonalelor pdtratice (funcllonale de elene
foma <x,Ax>), derlvata dlrectionall este: senna
o*(x)=t*-%
=(h,Axl+<x,Ah>= trrl t
= (h, AtO + ( A' h, x| = ( h t Ax> + I h, A* x) = ( h, (A+A' ) x)
spali
unde A* este operatortl adjunct al operatorulul A.
Gradlentul unel functj-onale pltratice generate de norna
operatorul A
depinde de x gi are.forna: llnla
Vf= {\,-a.+a.-*/l.r.:,
(84) 3.1 I

astfel incit: oot(x) =<fl,Vf (x)) (85) 1

O clasi mai generall de functlonale cuprinde functionalele care eleut
sunt comblnagil liniare de functionale llnlare gi ptrtratlce: spat
f-(x,;|url+(,x, g) , A-)u1Al S=)F;lg.l devil
(861
sir
o condltle necesard pentru ca x si fie extrem al unei astfel de
functionale este ca: lnte
Drf (xl =(lz, Vfz =0
(ez) exte
iden
adlci:
Prol
gg= (f,+A') **r=D Gftilx+p Sr=0 (88) rele

tn concluzi-e, constatdn ci extremlzarea gfuinpclElotnraatliecleorcocnadruecseulnat (r r
conblnatll linlare de funcgionale liniare
ecuagli llnlare cu nuclee autoadjuncte. Daci nu exi.stf, functionale tnt
llniare, ecuatille sunt onogene gi nu au solutie decdt in cazul in care
A+A* este slngulard. Dactr A este gi autoadjunct, atunci, i.ntr-rm punct 9i.
statlonar, f=0.
Ce.

TS T{sr.lo:{uTt{ TnT aTT{FXTTEOaUT 'euTp.ro u! 'lune TlPlTfEf}auT lnop eteC

x\r,\l(z6l st
lxl I | | r,.u > lxtr I l',t I : I

I

1S

(16) ( lxl ) ( I ?.r I* t; I ) > lxtr I + lxr;l : lxzs +xrs I = lx ( sJ + tJ/:)gIaape-r1u3

l

(6) rit'lr"itrl.ll{I rls't|J:li:z..iu;r,*"rlI

:ellfplayrdord acpJsTles roXerado Tntm PnIoN
EITuTJ'(ebluer?oTulupaugp^lE1In::oolleer.areddoo
'11u16.r9u :o1e:ado algamru as un

else I)

1=1,._[=._,tr,.&

aTTTteTar (

puglsTxe r-J, uTrd le?ou sJaAul .rolerado m a?TWp Fc caXclalrdord (
are TF relnbulsau al3amru as Te ATtJetTq eXsa I rolerado tm PoBO
A
(68) fgX tX=X1
(
elelTluopT
(
paalnullJrfnlolzzsloeondrdacuudoUoocaoedaalEspp:eau1(66!Iaa(n1pfc^peTgc?cleeJXbronauTuBoTTcupraJouoTlu€lrerTEargTedpupoJausgF(B^nueac)I:e)epulFeen.?ldra€TnucuoToncIud)TTengXalutTqraneauucotlo1T1ldef,3go1'Pp1ueunu:rauTa1Alxua1e1p3
q(t'u1)1o{eTrandTolETdosuTTeeJrlmsve'rsorTfTuJTolexpBraaldluoraedarantTm$dpudosc1€:Jeo1Tn1TarocleEJeaTdEoeTlunaTu{eedles E

u! r€TUTT nlfeds Tnun afelueue1a acnp JeTUTI role:ado m gc p1de4 (eg

JolrJolpJado u PlPWou Prqa6IH I't epi

'gJtcerpo eIPTuTr arq
€r[tou exEo u!
roTTr€uJofsupJl TnzEJ uS ec tgazpl a€uTEoaJlg(e[u'TEJ)aIp as lpurou Im€
I€TUTI nTtpds rm tSgsup pa lEulrou TETUTT nr{eda a?r!
't
un ad TtTrr[Jap TJ lod sJpc TJpTuTf rofTJolelado rornlnt Bau[l$tn TIJE
epruTaJo'1tan?1uunTT1ppdeJ,g,oeTBITT:rdTonlTes€ceAlld,eeoT?Tluenoez'lTeleopTTluenbpq! eTs'T::tT1sJceTTepBu:srTupngnTruq?vcu?leuonnde1sssen1Ideu1T{1cac1trnumT;1Tn;;TIa'rqpaoETlaeeTlllTuetreusuFuoeselTose&f
eo xolpbTTqo a?sa nfl 'U rdU esnT€uT TnTueuopoc TS TnTueuop pttgAB t<-t:I ETTC
alpTnoTlred are1u11 TrguuoJstrprl Tcap ltms TrpTuTf TTrolerado 'JBTUTT
role:ado a1€aunu as g=U arec nrluad g <- [:I prETuTI erElrofsuErX O PlIO

nvr0IDIM vtltvfi t[ ilriltxll zr-tT ffiItfiI ilo'I,vufldo '0 YIYAI

ullsts I$ ilt0cttc'flnlts

sHilil, cnc0ril $t flt$liltx 13 ,' [[gHnnT[ !x lrALIgA ruFcTtoflttA sBTIA

Schirartz. (e3) exP:
oPei
ln partlc'ular, caz

i[p n-.ur{*{p uo{*{p *,li* 1<p<o r-ot
pon
pn**'[pu,li*lf 1o1=1, (e4) vec
(e5) :'rul
pq
OP€
'o,l{*' CIL t

({,'.r'o*} *'(4n,o*; ; .{/ro"o*) *' 15p{e acl

r. r.Ir- \1 PTC
t='u,t - (e6) de<
JFaenlr4'dr*,}'[Ji*"',r*,; F
J,,
t/our utillea cu prec[dere formulele valabi]e pentm spa$ii ]iitrbert
Pe
r€sPectlv P=g=z. de

3,2 $ubspalii $i elenente invariante ale unui operator ba
liniar
ln spaSlul in care actloneazi un operator l"lniar T pot exlsta 3!
eubspatil $ pentru care xeS *> TxeS. tn particular, subspatille pot fi
unldlnensionale: elementele x (vectorl propril) ale acestor subspatii :l

sufertr sub actiunea operatorului T nunal o multtplicare cu o constant& -a

car6 se nunegte valoare proprlel- ata5ati vectorului proprlu x, adicf, s

are loc relatia:

Tx=Lx i9?)

Dln proprletatea de llnlarltate a operatorulul T rezulti ci orLce

vector proprlu nultlpllcat cu o constantl complexE oarecare este tot
vector proprlu corespunztrtor aceleiagi valorl propril.

Se poate vorbl astfel de subspaili .invarianfe ale speratorului 3n
cauzl: actiunea operatorului asupra unui vector propriu nu produce
ieglrea din spatiuJ. lnvariant. tn cazu] unor operatori care au un nare

nusir de vectori proprii liniar lndependen!,l se poate pune problena

nvalvoear'c6etaoErp6i tr€oppproroipeariabilrlel licnoarl,aduirnnuellinaddceeepnleeunialtdgiepi ntvlfcai;iltoianrtieparcnoe,perti,j"c,asz5-81ecosr:peuspnuendcEd

aTE TTJdord ro1lro1un alele6nfsoc xaTd$oc ?uns Xoun[pr7 TnTuoX€Jedo
a1e llrdord eTTroTp^ leco:d1ce: €zeq E?uTza.rdal loun[pr
1e 11rdo:d TTJoXoeA 1n1n:olptado
rBT lraqlT$ {nTf€ds lot pzpaJaua6
erel Ezeq o

aTnlTXsuoc TtJdoJd TTJolosn 'tTdwrs rrroxp:ado n:luad 'Bauauase eg
'aXIETTaIac e psndsue:1

ele$nluoc Erm ?uns lcun[pe znlac 'rS 1c'dJTp TnfnJo?e-rado arso?€ztmdseroc
(apr6a1u1 a1aeT3nu '1e:auo'*6("TVPUu-'.nrTE)F=)**ar{TarrJITIJ?Ptu €3 elErE aleod eg

'glsTxa gJPp 'Tg

'*vx*(8t-=V*)(={vr-(/**vv*)n='1*{=V-pI)
'-8+*q=*1gng 'U=*-V
:a111!u1ar coT np'T.ro?B.rado Top S TS H nc ulPlou p3po
H uTp (su8p)
nt{edsqns un ad T€ilrnu Jep (ETnuJoJ Tgeaaos nc) l3un[pe :o1e:ado tm
TnT eduaXslxa pJlsuouap e nrluad zBaTU TnT
TuTtap oleod as '1€n1o4 '*I
1E:uoa1f,eoa:a?dFo?Tes1oTO1I{i1lycuanlefpode Teur nir ToFO RTTqTsod a?sa TpuI nu T4TuTEJsuau
parezTle.rauag
:o1e:ado ap lnlnldaeuoc

Ezs{aose ss ar'c urrd (Tatr €T e?";;* lFTt:ritt#',ffi":" Il:i3l:i
a1{n1oau; glTunu (Frpun} aTlsrado JFcn}oxnu?onX?pTsueTTIaoprEg-s'THaJfnT?sueaoueopupI rgne
1{ru16-rgu
l9 reTuTT :olleoXeredo

(s6) 'EDi 'x6 ' 4;( ox*1g 7 = <4I ,x>

;e1{e1a.r aopJ:sTles aJpe H ed lTuT}ap (zsaTu TnT TauBroel pz€q

ad 'oTun) 1n:o1e:ado a?sa I TrlT Tnlcui'i[pp '*,f, 1n:oleredo 'a1{1u1yap
uTrd 'fo?erado ep TeJlcE tm I aT,{ 'a11qe-redas ?xsqTTH TTteds ad

TlTuTlap 1{;u16:gu TxeTuTT 1:olurado ET 'srunuT?uo3 uE 'u,r:e;a: eg

JotruJado $un Flcunlpu t't i

'e1e:1cads TszTTPtrP lPeTpap (st
1n1ox1dec uf nTfelap uI Tpil F1plpJ1 TJ BA eunTlsal{o ElsEaJq'TTrdoJd Ga
JoTTJo?caA €JdnsE 1{uan111suoc JofTJo?tsrado e1e (aTdtr.rs) rolltmlJce
pe:aunderdns trrJd ro€n €uT[[ta?ap eA as TET1TUT TnTuo?BJedo eaun;fce (s.s
u'glgayurn1d{ueodfs,saEpzaealeroaudoe6aTaTJrBdoJe.rrd€oTrroorl}ecraeAdJoofu,pnapcapTpB
Tre?ueraTa 1ro1e:ado f l l!,j
'a1red FlTp ap ad

' tTdwts ytoXetado fiTunu
luis ellface :g p1{eds 10? pzeeJaua6:o1e:edo Tn[m Te TTrdord TTJo?3a^
€J€J uT eTa3p a?$a T'fqeJoAeJ T€ur Tac Tnzp3 '(Ero?FacB
reop equtqos BA Tuo?EJedo) 11-rdo:d Trolcal ap elJeulqnoo eTTJspuod

o sc 1o1

ewg"rdxa eA as H uTp ?usuroTe Tn:poTf,o B:dnsp In1n:o1e:ado EaunrtcP 'zec
?sace uJ 'TTJdoJd TJo?caA ep gJeTuTT elJeulquoc o EO lep lnrolerado
ezeauol{ce aJEe uT. (6arXu3) TnTnTteds JoTe?uauaTa roJnln? 11:gugrdxa

YlYr0lm{flt vztlytY t0 iltnnt rz"[T ullsls Is rullc{ll 'ilYillls

$ntrln, cltfilIil $I $t8nn l-3-zs H,ilfinr lB iltilut tutctl0illt tEr
op€ratorulul dlrect. Hll

3.1 @ratori autoadjuncti gi operatori pzitivi H.
@eratorll pentru care T=T* s€ nunesc operatorl autoadJuncti, Un
cp€rator T €ste a(uxto,Tard)juengcttegrlenailrogrlinciatredaacri gi mrnai su
daci
fi xeH. sul
@ratoril nen{rglnlfl pentru care T=T* (pe domenii restrAnse din
8) ae nuDssc de aeoenea autoadjuncti. 1.
Operatorll autoadJuncgl au vectorl propril ortogonall, valori
proprll reale.lar natricel.e (nucleele integrale) care le reprezlnti .d€
xurt hernltlce adici lnvarlante la conjugare gl transpunere.
J(f-f*), I rr
Operatori autoadJuncti sunt (pentru T oarecere) T+T-, TT-,
.p
?*T, r'
Conditia necesarf, gi sufici-enttr ca t sd fle autoadjiurct este ca 3d
frnclionala pEtratlcd <Tx,x> sd fle real5 Vx€H.
dactr gi nunai dac{ tlu
Un op€rator autoadJunct ae nrnegte pozltlv cF
<x,Tx>>O Vx€H. Tt* 9l T*T sunt pozitlvi.
F
3.5 0peratori izoletrici gi unitari
un operator v este lzonetric daci I vx! I d , VxeD, unde D.,, este T
dcmenlul lul V. Operatoril lzonetrlcl lasI produeul. scalar lnvariant 3I
gl reclproc, orlce operator care lasi produsul scalar lnvarlant este
lzonetric. f:

lntr-adevf,r, &
:r
a. Daci (Vx,Sr>=<x,y), Vx,y atunci, evldent lvd {:d , ft

b. Daci V este izonetrlc, aven, pentru z=x+y.

laf2 = fxf2 + I ylz +2 Re<x, y> (ee)

c1 {100)
( 1o1l
llz(x+y; lz=(V(x+y), V(x*y) y=l;rl2+lylz+ZRe<Vx, W7

Crn V eete lzometric trebuie si aven:

Re(Vx, Ux>-Re<x, y>

Lutnd z=x-jy, se arati analog ei ( 1o2)

Im<Ux, Vy> =Im<x, y>

gi prln urnare:

<Vx , V\>. <x , ys ( 1031

Un operator lzoretrlc avAnd dcmeniul de deflnltle tot gpaLiul H se
nunegte operator unitar. l,lultinea operatorllor unitarl peste in spatlu

Tf eTt BB -I"?Tun rolprdo 'f="n=a-fil PoTpe r-n=*O=O al8? lcmfppoXnB
rm €c pluBT3Ttrns Tg Frp8eJeu €T{TpuoC
'Tlcun[pPolnE luns D'g't eprm
(c[)dxa=n ne8 r-{s[-r) (et+r)=n',-(r[-u) (r[+t)=o

:PW:tOJi

€p TTdBIsr FlsTxa Tl,3un[ppoln€ Ter T€ TrelTun TTrolersdo ejluf
'eTeuJououo azeq elre uT exgflJouopo eTszeq eiJotsrrpJl

?od es s.rea ufJd TnilsTueoeu pzeezfuml Trelilm rTJo?eredo ,TeJTsV

I (-i;
{OII } o-<:rRi,cFTSpp>-{"<t"*r7(,1F"Q"n}> o-<rO,r0>
(ITqErFunu) lalduoc
lpuJouolJo
le8 xtP u3 {""'(0"..} (ITq€rPunu) lsTdEoc lpfxouolro las arTxo
rEmlurFolJsrrup1Jgl .reeoeapleToedunaugTg.TgTcBstp[JJoEuolTUuon alse XJoqTTH nTlpds un ed rolerado

11rdo-rd TTxoXceA ere ,rp1ncTXred

? 1e{3gtT'p€rOc'e^'p)f=rT€F'XeuTFrrrmoFo-anrlosI?aBTgJ6B=-dnnoq=Itpn-uITlTu9gT3u}oF'pTlTlfnlrzJeOTJrxleeopElemoJzeTadpoall=us*neffei=rTpOz?*nTfllJmupotsJcpoulETIrEeudmou

{60r) t-{7=rO (='I=nrlf E

:POTPE ;

(sor) qi' x7 S - qA' xn"n> = <ttn' xn> = <r{, x>

ep Plc6aT 61Ba :P?fnEeI'<rl'xy=<r(11'xn> ooereosg'pun[pe lnrolerado
plTqasoep elelelrdor.dl=O
TrBlTrrn JoTTJo?sredo B

,aToerd Tat{.lTarTbrpu TF cTr?,alrozT aXse t_n 16 pc plpzar it_fl i
eptm ap

tror ) ' lr{ = fx6l = lxl = lx;'l = lx6.-nl = f,f1-f I F]

:neAg 'xfl=A pugund

{eor I I.vl1l7=I|r-rt '-t
Tg TTqEsreAuT e?se n Tcap H TnTlBdE 6er1u5 alsa r_n TnI fnTuaEoq

(sor) Hatt'xA' <,t'x>= </rn t xn> P1u

:JPfEcs TnsnpoJd gze;reaur 1$ i-ro
uIp
rlt(tgl)
un
: (I e16e
fasrou l1-lpl <= 'lxl=lxnl 'I

€TtrTuTJap uTp X) 1{1ut6-rgu lrms TrptTun 11ro1e:ad9

rutsrrxn usrruorrudo rll.r,eruau6

'dnrbqns

3c apnIJuT IaI palrnp1andand:6leueotree:ueazur1ppeT:eJul ar,6iETopTtTs3TnnalTflnsTuloeodednu:T6p TTrleurys
1€
11fqo: lsact 'g
elsed re11un lndnr6 lTunu TJpnJolsupJl ap dn-r6 rm pzeauJoJ H ?reqTTH

nTl0llct0f vtrfi[v tr rl,Imlx 9'-ET yIruoi

Utxtsls Is [Il0tutc 'nvfifl$

$BiIil,B, CI|CUIIB $I SI$nilB L3-zt BtHlfix n ililtzt rutfiI0lAlr 881il1

Do1 operatorl T, gi T= se nurnesc unltar echivalenti daci existi un Fls
operator unitar u astfel inc&t:
N=rl
T.?,-.=1-f r-!T lf (1r1) 2.

3,6 0peratori sirpli gi operatori nornali vaI

Un operator este siaplu dacd vectorli s6i proprii pot constitui o ace
bazX pentru gpatiul Hilbert in care actloneazi operatorul. sun

Un operator este nomaL dactr sattsface relatia: val

I rd { N*d , x€H. Ave

conditia necesari Si suficlent6 ca un operator sd fie nornal este Ei
ca el sl coslute cu adjunctul siu: NN*=N*N. tn acest caz existE
relatiila:

Iaal { A4 { Al =, IA'l { el-.

DacE A gi B sunt normaLl atunci A*, lA, A+8, A- sunt normali.
orlce operator nornal ge poate scrie sub forna Ar+jAa cu A1 gi
gi reciproc. De asenenea, A2
autoadJuncll gi comutatlvi lar A*=Ar-jA= un
operator nsrmal ee scrie unic i.n forna N=HU cu ll pozltlv gi U unltar,
conutativl.

1. 0fsgnvgf:s Exlsti nulli operatorl normall particulari
rennari:.:hrj,1i, V, care satlsfac relatla <Vx,Vy>=34;,y) unde K este
ounclotanrslt.gani tpi.oAt cfiegttrlaotpaetiraintotrri-udnifuelrEdinfouarrnteitozaurteinle
de operatorii

modurj-:

a. Operatc':iu1, nodiflcat prin implr!.ire c\ Kr/z, devlne unitar;

b. Produsrri scalar se redeflnegte innultlt cu factorul 1/K;
c. Produsul ncalar se plstreazd, dar baza se considerf, doar
ortogonali, nu gi ortonormati: 0r=gq. sau r$r-K0r, 0. g1 0, ftiJrd
baza dLrectE respectiv reciprocE, astfeL lncit <0*,Qr)=6r:.

Inportanga deosebitE a operatorllor nornali constE i.n aceea cd el
sunt singrur:i" oparatorl ai cdror vectori proprii (invariantil sunt
ortagonaTi. Doui categorll remarcabile de operatorl normali
operatarii autoadjwncti gi operatorii unitari. surt

"T1:1?T$n TeE ep fS f{cmfpeo?np TTrolexado ep

':eTmT?JPd uI 'axnrPJsT?Es luns 1{g1e1rdo:d elrecP '"mETEeq .r

{flr } 4*{{xix'rfxf srrrxy=l-< fx , IxiX } = 4 rx , rx* lgy =
{zlr} qtvtry 'TN>=
<dx'tx>{\

1s

r34.ry*rx*AI { - rar,x.,ixl't}=<rx,t*>ty=

= 4ryr1 , rX),., {IXit/ ,I51'> = (136 , IX*/V>

: ux6nv

'sx Tt€1*-r'x1;11"srdog":dlemJt)x{o}"uroJ!3*6?rp1laetl:.e.JolmgelcmndFsJtsspr'orsou0s;xlFurTslspg'p"rp1nga:dpoe..r-d:1uT3:oTpA
gnop

'. aTTBuob0XJo ?uns
313ut1sTp TT.rd*:d TroTEiA rourr r.r(rlezirndsaroe 11-rdo.:d TTrolJeA -

luns TTxdord llroxraj\ .r€1 i'l.laJ1{l' 1n;nrr",1erat{{r 1B l$elaeoe
eTE tTrdoJd :ofTroTpA
slalebn[uoe Suns lJli?r[pH g:r1n.:*1e:arlo 6t-e ;i:do:rd €I"[JoT€A
rIuFtrOfl KrylT$&ruuxdo rln LlvJ.lrudoud "z

"oXstJT?BTp TTJdo.rd floTpA {!;}:ail '{ ITirT:,rlre!-rdIpspr]u?n!s ] rrTlTifi.tot{=}l' {{€trffiorr=ffi
iOl r-ro1erad6 Z't'bTt
';1drgs=g : a1;qc.recl+* '+isqTTH

";

,{ I
I

d

nn0trlrM rutiltl fif, stfi!#rt* 8#-fi [ il.iflr$t5 i$ ruifitui3 'l1Yfll|a$ i1l

$lilEttr, crafutH $i $i$!$l{t t3-r' illililBtxH[tT[ B[ ruI[TI0IttA $iltmtB

de unde rezultS: ( 114) op€
ll=H
(lj-It) (x;;X.)0r * (x.,x.)-0 ini
llp
3. 0rsanvnfrg operatoril normali sunt nai generall dec6t cel
autoadjuncti prj.n faptuJ- cX, spre deoseblre de cei autoadjuncti, Alg
valorj"le proprli nu aunt cu necesitate reale degl vectorii proprii oper
sunt, in ambeie cazuri, ortogonali.
!a+
3,? Analogi* lutre operatorii liniari gi nunerele conplexe
tntre cperetorij" llniarl $1 numerele conplexe se pot stablll o lal
serie de analcgij". Departe de a fi perfecte (complexitatea operatorilor
nu $e poate ecnpara cu cea a nunerelor complexe), aceste analogli pot lAl
consti+"ul o mietsei$ de a flxa nai btne o gerj.e de concepte legate de

operatorl.
evj.dePnreiEzeontcaorereas6alnonadloenglidilodreotipvo*nI'fd,aiccteioinnatar"beinlutrleurdmiftretorlrtecarteippuurni.e in
operatori gi c3.ase de nunere complexe.
de

Re.marcEm* 1n speclal, acneaalodgiinatrdeinotrpeeroaptoerriai tuonrilltaauritogaidjnuunmcetireFlel

nr+merele reaJ.e precum gi
conplexe situate pe cercuJ- unitate din planul conplex.

nunert

este.
dlviz:
lucru

'ro11ro1eredo erqe6T" rrI ler9^epu'1eraue6 u1 'e1sa nu e.rpc nrcnl
nn(TucTrgEopTrsqTraaJAbouTleTp:eeodlsosee?Trsqnoue6eTxpnafneldlcuuedouwcogt.erlouTe!elfrpdaAuemTcxlupalnBunrqoceclEuaaT?upsree'TpaeaxueeaolduTuarosocp)
eunTzT4Tp

aC 'olsa

rofexaunu
u:qe61e 'aTalIE arluI 'elceg:ad lrma nu ofTT6oTpup Rc rn6Tsaq Er

?xsTduoc Eerebntuo3'e1{n1oau1 lu F'Fl-ViiJvIalP'sI=l'OwF< ilI'vF'0IF='VFlv<Fl=le>Fl=0+=pvleViI a
re1'1n1npou prITJ Ecrou : JoTTrolpJad'o
IS
'erelduoc rofexgmru eeqs6lV e alfnlonuT nt Bler&rou erqablg
IF
J=11-ty16,-nlosnosl Q'OlZ<l=lzzl xBlTun O 'nTlTzod t{
z exaTduoo eresnN ' { TnTlPlnuoc) 11'1=1114=g uI

(aXeXTun Tncroc) I oJECAIpO IPULTOU :Ole:ado ap
10
I 1npou ep eralduoc roleJ TrE?Tun :o11:o1e:ado lndn:g
fo
-EUTU TB ^T1€CTTdTITNU TNdnJg 1{c,,m-(frpte-3ol)np(r['+tc'g)='vn
oi
!(la'clel/el[t+'cc)'q='z$ nPs r-(g[-I) (s[+r)=n
ax
nPs {q[-r)/(41+tl=z nps (Vt)dxa=6
TTJ.
nPs (0[)dxa=z I= p I 'n TJ€?Tun 1:oleradg 'T1:
TAJ
I=lzl 'z axaldloc aJeunfl gcs T^TlTzod (T1run[pEolne)
TreTuTI T:olerado (r rr
0<x
-H=V flrru
an111zod (eTEer) axetrnN
lfcun[peo1n€ TreTuTf TJolPxado
eTPeI gra-8lt=ruXl er(T^TlBlruroceu) TrPIT{m'O'n

I=l(0[)drel '1n111zod '-n 't{ '.n.n=n{=V
'Tpor 0 '0<r '(0f)dxa'r=z
nes 1{crm[peolne .{,'x '^[[+X=v
nes TIPoJ d'x 'Al+x=z V aJE3ereo TTETUTI TJo?eredo

z erelduroc sJamrg

YTYtllt$t0t tznvlY ff ilrxrfll 0€-gI ilrusts I$ flI0iltc 'xlYtrx$

$}ilillB, CIteuiIt $t $I$lHtl ]-4-t flr,BilItIE DX tXil,IZt $PBCtttil $Bi I

ELEMENTE DE ANALIZA SPECTRALA se

OPERATORI DE PRATECTTE, REDTJCTIBILITATE 2 dis
2
Operatori de proieclie 91
4 g1
RaductihiTitate operatorilor gi spatii invarjante
6 ac
OPERATORI TN COI{TE]ff SPECTR,AL 6
t,
Dlade 12
1.1
Ac|,lunea unui operator liniar gi ndrginit intr-un spatiu 14
15 1.
HiJbert (bazE ortonornatd) 1.7 de
20 e1t
llorea unui operatar definit pe un spaf,iu Hijhert 20
21 xa
separabiT lr.rJe 22
24 pal
Exprinarea sennaLeTor gi operatoril.or sub tornd de
pr(
inteqraTa A,

Fanilii de operatori -1,

Scara specttald H(

DTSINIBUTTT B

Generalit{f,i st,

spatiuL S de funcgii test i{i
Operatori W spatii de distributii
de
FORIIALTSMW DTRAC <BRA.RET> 2.

Analiza spectrali este un capitol de anallzi funclionali gi are 2
drept scop studiul nodului de acliwre a operatoriror pe diferite spagil
llniare netrlce (topologice). fdeea de bazd este aceea de a descompune, T:
oianptvueanrrcaiaitoncarteind,niaenls"tpesalpnrotpicsluiiblai(lp,r ruuonnleidoclpnteoerrnai)stoi.rcoaninraetlre-a,ocalsiloeunnesiapszaalullupinleutelsigunbri!as1plaia1tdi.el P(
Operatorii" de p::oiectie vor reprezenta, astfel, o clasi importanti de
oapoepriaentroacretlorlereilcgoear tuairumdteeoaadzdedsjucnnoemcvtpoiurngneirreeufaneslrtelamcrinuaipnrerelsocrpidianelrieiseHlnainlbateleeoretrllaesenspepanrteaacrbtelrt.ael.r
Ilotivele sunt legate de apricaiiile lnediate in teoria semnalelor gi
de poslbllltatea creerli unor
eistenelor gl analogli in vederia
inielegerii princj.pillor valabile in cazurl mai generale.
Pentru flxarea ideilor sd renarc;n ci spatilre Hlrbert separabile
Bunt generalizarea inediat6 a spafiilor cu un numir flnit de dfurensiunl
oppnlnpluretoerc.reBsaaurrtencolitarpiirienenrnesstncripuaoparnaecttniaeruafbxleltoiceluigtuleiacis.llilpnEdcatixiaurtllnoisutritltvuirvieiode.HicntAteielos"onbralrseicipe:ou:mrtn.oaaapplnverrienel dcapntelruoearaazirlpldiotna[trtarilgtipu,ninaasvrnptpaaemarlo,occrgisultiruiloludnerelt
llilbert pe care sunl definlti. Tratarea riguroasd a acestor situalli

es x TnI p.IdrrsE Jat nc lt+d TnlNolpredo TnlnsraAuT BatmT:atcTEJJzsaalr?r=cdx l

r€T'tx=(x)d Tg eTlceTord ap JolPredo alse d pcpp Rc FzBarl6uouep as a

'I5dr0 I
'<86'x7=" l(x)d I
P
P,r Ir l(x)d I
T'
:ei11{e1a: cBJgT?EE Tg 'T^T?Tzod Tg (d=zd.) TiueaodmpT
a
'(*d=d) TlcunFpEolnp 1:o1e:ado lrms TTrolceTord rlF$udoud 's T
E
uTcTT{ol TJolcaTord
,e
?uns H TnT eTB eTEuo6o?roBu TTledsqns utp eTEs elp alueudroc
u! H trtp xuerlsTE rm cnp €xE3 6T{,ceTord ep TTJolPrado xltrAus6ls0 'z ZZ
TZ
'sxeAuT TS nTiedsgns un tTun pou u1 lpTcosp aXse Tg a1{calord ap
ITonlpTxneTdlodsrnauITnT:lTETalIEsqdnsqsnas relulsgI oeETeardn?aeTut)aeT{uoaJpdueopdsTaT"lJooclETreedtmo oz
uf TS UaqITH
earTITqElE OZ

lgurad pc 1n1deg Alsuoo eT{ceToJd ep JoTTroleredo ealelTlTln !t

TTn6T(lE'adsTn'TTeTJnlsTVue'utopT)nT! 'gTg nr 80U=ll : ('d TnT TE Tnu Inrleds) S 9T
roTTTlBdsqns e (FTBuo6o1ro) F?carTp Euns alss ll
,I
11uuo6o1ro JoTTrolJeA f,oJn?nl eaur.r{1nu uTp poTpp
I TnT TE I TEuo6olxo Tn?uaFlduos uTp elJpd scEJ tor-dx=rez1l'(n1luae1r{creeTpgord'v ZT
algeunu os
ed :o1celord nes eTtcaToxd ap rolerado 9
errrroeX) RcTm eXsa t TnTiedsqns pT pATleTeJ x TnT sa:aunduoceaq 9
e1ccueol6rot1:fon'gle=ul(esxea)rordlEe'cvBe6pluln'Tlur^osnploeTreAelXdTop'HlneTelncTo'gnsTeefeddasl1sepeuoTn6TIolp1eTrdonsJgapnlc5segunz1-nral1upnTu1p{nedazs+lrqr{en,-dsx
e1{e1er ugrd 'E ad x 1n1 pTlroToJd lTunu 'gAA luaurcTe rm Hax ?uauaTe ,
TnIPoaTI €zpTcose H3ts<-H:Yd 'HrE rnTtrBdsgns ad eleuo6o1:o aTlsaTord ap
n1rTr-terod1seruandgd'a1T{trcTaulFolradp o Z
ep roTTrolerado €s€Tc sTnlTlsuoc H lJaqTTH 'I c

1:oleredo ep Eluegodrr;E psplo o 11.|'111dSg Itt

I'la1{carord ap IJolPJadg

ffi[,l,ITIgIIJn0XU 'U,bU0Ud fl0 IUo,wUld0 'l

'1n1n1o11dec e
apToTlxToidrquT.!rtaellpp?upargzaarld:dluanldsac€uJooclsapc^ealT€cTr'Facls1J6lo1Ieeuuels:ToT1eauureo1Jsu1sn
FIEUTJ TS
13
urrard alsa rolllfnqTrlsTp

roTeTeuels ETroeX nc FrnlgbaT u_r EJpseoeu lnTosqe uf eceJ aleod es

esrErepTsuoc 'fatlev '1euo1{nqg:1s1p lxsluot

trtlnt{$ tztrfrf tt trffflilx z-rI flffit$ts I$ ilI0iln 'flIiltl$

$ilil[x, cltclltlr $t $IsIl[[ 14-. ![ffIE}ITIT[ ATAI,IZA $TICTTAI,A

(P-Ar) -t*= 1'x-tA- - $ (x) $BiltttB,

A 1"2 Re
FI
Li-nlLa orlcirui gtr monoton de prolectori este tot prolector.
ltulffurea proiectorilor poate fl pusi in corespondenld blunivocx cu gi fle
nulti.nea operatcrilor unitari autoadjunc!,l prin relalia:
txeA.
unde u este untar autoaajunJt,?J;'::ll/jp"ruto*r unlrare. (Evident, operat
dacE U*U=Uz=I atunci F=p-=pz gi
DacA P este prolector/ f-p reclproc. ) DacX p Si 0 sunt Da

este tot proiector" estg u
prolectorj. Fi P+p este tot prolector, atunci pQ=Qp=o"
ln spatiile Hilbert separabire, operatorii de proieclle ortogona]"i T, at
sunt reprezentati de clasa matrlcelor dlagonale, avind pe diaEonal.e restri
nunai elemente egale cu 0 sau 1 (gl evident toate elementele se afl
nedLagonale nule),
D€
l. Ermmr,n PrezentSur operatoril de proieclle ortogonalE din spaliul
euclldian tridJ-nensional R'. Reanlntln cd acest spaliu este spagiu cEAI
Hilbert in raport cu produsul scalar
lreduc
(F, Y> = (7x. +J>q +kx, ,avfXyz+ky: ) = (21
1. Rr
xxy-a+-'l.2f2+xxf7t unde
nl
E=3xr+J:rr+kx3; V=Tyrnjyr*kyu; subspi
subspi
:L,i,k, sunt versorii triedrulul ortogonaL de referintd Si satisfac
relafiile I ob9lni

{?,7} <7. T> =(-J7.t n) =(7.'JF) =14 r (3) It--l'
=<T, k) i
*{7, T> J. O'ne,.
=(k, J} =(J,X) Da. se
=(8,J2 =p dacE

vectgrul x admj.te deci descompunerea: {4} connpol

F= x.7, Nr7+ x_E z. I
r
ln acest caz existd urmitoarele posibilltdli de proieclle a f
vectorulu"l x pe diferite subspatii:
FrF=Txr; Pr*=jx"i ep=Fexr;
Prr7=i.xr.+jx"; P71p=7x"+Ftxri ps!=Txr+k:cr. 't, JE r

astfel incat vectorul poate fi exprinrat ln funclie de diferlli

prolectori {n uroOut unnitor:
'-x--pI--.7+-Jp"' .*+P.T=p.E+p,E=p..E+r-!=p-.;*p,1= (O)

ep TTqpJPunu lTuTJuI nPB ITuTJ rPunu m pUEAP elBuoSPTp nB8

erelnppunTrx JoTxdns eJTr?Pu :e?PTpa6F xuns slEuoTsosutp lTUtJuT
roTTTids fnzBc EI ATFnTsu[ 'TTlpds alTna TEU pT aTTrpzTTsJaueg

(or) ltal l*loll - (e
Ilot
ttn'] :SCJOJ T1T.

qns aTJcs es TT{sdsqns Pnop ep fTqTpnpat TeTdwoc rolerado un JeT

(6) ltat totl _ (s)

It*t to] l=* : PutxoJ

)i gp aJTr?,Err o ap lpluazeJdar alsa lltedsqns Pnop ep ITqT?cnpsr Pa'
rolpJado [m 'altuoTeua[rfp ?TUTJ -ro111fuds Tnz€c uI fndlNrxg 'z
(r)

ou*T( TI6gep)tJrxTpJntlTxueeetp1nTJaepTTtmupeirlNu*To1cnpeu, npETrrEScBrd''nUgn,es(exT! nIgeTacdveTe*1pnt1uT'auaIrt1*u:mv1o\vTal;venpfacJTu=a1e5tdnp<orp*!o,axc1sucan'elpmTT1rfSmocT:erldogTeEJSBcs?t'ssaueTulrJ!rgcod's'JTx)d€e(pgl'e'dlgutoet-3duIx)o'.eId!ugrsucF*roetIOcp (s)

13 '4*1,:'& :1frgd lnop E Fpe:Tp tuns axsa J, Tn:olpredo EJ es-npugufdqo 3€JS

{ [] kr6o.lo.u (z)

:eTr3s lcep aleod eS 'I TnT eTe oluETrBAuT TT{Bdsqns nr{e
luns 't TnT Inluamalduoc '\ Ts 19c ! lBxp lgrul laIls€ g nTledsqns TN}1
{m ep lnslnupa€urtoap:}nsecg: c.s-p TaTdwoc al9eunu
ae "T:pr@rrouleT:SedFocpup nTTE(nITpIn{Hpo€Jr 'I aralI
fu l"f,tIIIIuJ,c(EU
aT?uc
'TTqT?3nps:T TTPUC
n€oeeJll8esse3efTJJeq,, TJ1rppnpr1fnX,pueuaoatr31,':oaezl=dese")'C,LuI ITepmd'rtua'reuupnodAslsnn'arptosalurealaIrueXe'*dsT0eoJEIPTacur'TggTTcId1ueP1gpfeenedrep?ssoBeoIJldT, au1Bdpn'[Ee1a$oueTn1IcPp{rPccmap1Trq?l]1tBasaBe',r8c[ ?uns
nc Uoder uT lrlpTrp^uT a?se f TS PftOcETppnclTupueTetmrJoEdpAsurcTesaHlzsuEa!ctrol'sIl'Heer=Tc€enEdqTonuFrstIometqe'UroelasdxeoI
'luep
..! gJpq 'Tfrisug Te uT E ad ecnp &
IIC €Cr
fp lueTrE^uf nTipdsgns rm else
Tilml€ 111t3x pepp po palpleTrdord pugae H TnT Is nTfpdsqna m I aTI TS (r)
H:qq Tntueuop pugAp H lraqTTH TnTteds u! rErlTqrp roleredo tm I eTJ

alrnlrpAuT TTluds 1$ rollrolsJedo Ealeuil$pnpeu u"l

rlflmtrs wrlvtt x[ xrltfltfi ,-,7 tttlsls Is ilIncuc 'flIilIt$

VTYII:

$ilttlB, cIlculTB $l $l$IBIB 14-s IrltrtItfl DB tttllzl $PxcImll sBlt I

elenente. Constat6s ei reductibilltatea asigur5 o antrnitd Pot
!'decupl.aret' (pariiall sau totali, in cazul reductibilitigli
couplete) a relatlllor care definesc actiunea operatorulul. und

3. Vrc'rmr gr vBrom PRoPmr sX ne inchlpuln urmitoarea Hslgtuiaeliset:e var
operatorul separabll
T actloneaz6 in spatiul Hilbert cAt,
redus de subspatiul ArcH care are o singur[ dlmenslune. Prln urnare,
,)
actiunea lui T asupra unui elenent din A, duce tot Ia un elenent din
Aa. clm insi Aa este unidlnenslonal, rezultl ci elementul Tx, L,
corespunzitor elenentului xr€A, este proporlional" cu x', cici toate 2,1
deelemuEnenvsltdleneleEnrtdrdlnealecArin,xesrnu=tn)3.t;Ard,ceTefaxosertm=aTaianaxsxae,=anlcrnauixurc=nil,xxr.tpela{nurtbrriurittrrxLa,€r(4c(1os,unTPntxlergx=e)nlrIexrr'a' tsee pal
ntrneste valoare proprie., iar orj.ce vecton din A1 se nunegte vector
proprlu al lui T corespunz{tor valorii proprii lr" observdm c6, pentru H:
un ele$€nt arbltrar x dln H, conponenta care se afld in A' este
proieclia lul x pe A, deci relalia corespunzitoare vectorilor proprii ar
se scrie; b

TP*rx=lrPoax, xgH" sP,

Reanlntl"m cI operatoril ai c5ror veetori propril genereaz{ tot 1.
spatiul Hllbert srmt operatoril sfuipli,iar operatorii stulpIi. al ciror
vectori proprll sunt ortogonall sunt operatoril nornall t! Xd i N*il sau J!
Nll*=N*N) intre care operatoril autoadjunctl sunt operatorj" normall

partlcular1 .

o cl"asX particulari de operatorl simpli este cLasa operaiorllor

degeneraSi care au un nunXr finit de vectori Si valorl proprli, spatlul
Hilbert pe car€ sunt deftnitt fiind finlt dimensional - eventual un

subspatiu af unul spatiu Hilbert.
Actlunea unui astfel de operator se poate exprirne ca sun6 a

actiunllor oparatorllor de prolecti.e Pe cele n "dlrectilr' ale

spagiului, ponderate cu valorile Proprli corespunzitoare vectorilor
proprll care genereazi spatiul':

*?-*-E rrrrx=fr rror0r=* Srlr{o' x), "r-t. ( 11)

Generallzarea la cazul lnf lnlt dj.nenslonal corespunde operatorll"or

nornali compacti pentru care;

resperctSlvembanzlfelci arellic1leproncoetaggilillonr:a0ce" lasgi i S. sunt elementel-e bazei
ti-nrp vectorll pr rprii al
operatorulul T respectlv ai operatorulul adjuttcl(v" nota 2 de subsotr),
l. sunt valorile propril ale operatorului T, iflr' Q", spect,rttl
elemantului x ln raport cu baza {0"}.

p'xunTsnoTnplucau{*a0Te} :PaxelToAzap ,TBIXSV

ardnsp apETp ;oun €aunTlJp uTp alqTnzar a?uauoduo3 ap
F?Ep EzEq o nr lrodsr uI FaqTTH nTteds Tnun r€ ?uaur€To
Tnrm eaJaunduocsep elerdreluT ura?nd ppeTp ap Tnldacuoc puBzTTTln

' '(r'z'zl uI olp?ncsTp rJ

roA af,E3 aTTTlofr?se: nc 'aueolm T€ TTUTT ap ?TuTJuT Jpunu un sre
€acTJleu 'a11qe:edes aTeuoTsusuqp lTuTJuT ro111{eds fnzee uT JsT

x{.* ::: .*f' .ql I

(rr) Ilsq'n"""''' l-o- '' 'o

9"e Jq"l
fq'n Jq"l

'aleuotsuaurgp XTUTI :paJTJleu ep sTJcsap alge lnJolpJodo .I t

XraqTTH roTrTipds Tnzsr uI flIIU USSSg ''

'€ TnJolJeA ap Xsreuao pr{eds :fEuoTsueurypTm nTtBds !
un-:lu! ?reqITH TnTfEds ?o1 guroJsuerf ppETp O 'x fn.Iolcaa 15 q
1
Tnf,oloaA eJ?uTp r€Tecs lnsnpord pUTTJ alpXTleuol{,rodord ap tn.rolceJ ,B
r
UIaaJqoT?TrHaATnnTr4BfpdusouTTf,proxdorord?craoAlcuenAEru.lm.ro-JrXsrurp!.rTl JoerpeaE<q>x<'qE>lexXolocuaA,reu1nn-cJ;?1u.rgedH g

agtcerotd ap roleredo rm alsa (Troloa^ rop B uro?xa snpord) pp€Tp O I{
"f
apsT0 I'u
a
Tvtmfids ,I,xffi]lol ffi l[o,luufld0 'z
atr
'roTralIn 3er6ered un-Jlul 11{eraplsuoc pnalgc
are} :oA aB aree af,dsap 1B p1e:1cads grecs FlTunu X FTeeJ ?TTqBTreA u
ep pugzuldap aTtrcaTord ap 1ro1e:ado ap aTTTurpJ o p?uTzaJdeJ (T)d epm
,
{€1) \ttu/ot P,aIu,J=f -&, a
:a
4
TT:
: Eurrol: ug etrp:dxa 1od 91
as TrB?Tun nes Tlluntpeo?nE JBp'lfceduocau lT-roXEredo '?TgJgJs uI

r-r Tr- T.r

${.zr ) 'r=tu , <x,t0>tYt0 f,=rgrlrY f,=xr6,rx 3=xz

tlfutlds v?tIYff [0 xxlltfiIl e-rrT iltusts I$ [IIflcilc 'fllvtlt$

$Blll[x, cncullx $t $t$tlilx L4-, Bl,lillfit D[ lililzt $Plcrilil $Bill t
adju
x=p 0r>(01,x) (1s)
sen
poate fl interpretati ca rezult6nd dln multiplicarea la stAnga a lul
Re$a
x cu operatorul identitate scris aetfel:
2,2
r=f, 0r> <g., (15)
Hil
pilberbaas-zkeaectc"oinr(dtsieatsaptrrreeclcadalrieeasstaeendtaaevuralconatataerJvioasaeasnis6ucnrtullenlit:zeaIraesafdforgrrintualllscmapuiltuoiluDllurai)c
su6
(17)
tral
elementul xtrbfrlian'rdu'fuldnnea-Znedm"en! t).nkeRte"n(acractrrem [agteapti" acliunea unui
asocierea setr
operator componentelor scr.
vectorilor a> (,'ketu; $1 <b (ilbra") cu coloanele respectiv linlile
dladei. est

Un operator de tlp diadi de forma Or><0, reprezinti lordpee-raatluonrugluldne de
proiectie pe spatJul (unidlclensional) generat de O, Dac
conplementulul siu (neortogonal) generat de mullimea {ok}k,. gi av6nd
drept factor de proportionaritate, valoarea componentei pe direclla o* adi
a elementului proiectat. 1ar

Dacx t., reprezintd natricea unei transformiri linlare T relativl V€r

la baza O*, atunci reprezeFnTtare(a0d,)ia(dt icrjijear)operatonrlul este: (18) de

lntr-adevir, S*=T rr*Or=F r0r>(? tif)o*=p tr^.g, (rgl

adici actiunea operatorului asupra oricdrui elenent al bazel este
aceeagi.

tn cazul in care baza Q" (neortogonali) este reprezentati de chiar
vectorll propril ai proprli
operatorului A, presupus a avea valorl
distincte*,r=1{"8, l0a-c1li)u(n0e1a,a*rc,e|s=\tuEla-AaOsui>pr(a0uj,n*url=eElemlernQt xr>se<p0ora,txe)sc(r2ie0:)

0. ftlna vectoril bazel reciproce (vectoril proprii ai operrtorulul

gcep

irgry=<t0,F0>iT=<i0,r0J1>'16*r1=rg*g
Fr gFy=4Fg rrgyFl_ aFqf1 rrg)= <FQy,t0>= <f0,t0, F>

aleod as 'pcordlcar ezeq alpadsar 'v rTgnlTnSrorlOprFaOdBopr'eJpA11aipd€"r-urlutltizSllSl

uTp pleuJoJ 'e1carlp Ezeq s?uTzerdar

'TT{p?Tun e eTjnTozel EXTunu pgp aTnlTXsuoc arer aTtpTaJ

{sz} '" 3=,

:e1euo6o1:o a1{ca1o:d ap roTTxole:ado euns olsa €?plTrm 1n:o1u:ado re1
'r0 ap aXpuTurraXap aTTTtpdsqns ad aTEs ro111{caloJd Bmls else x FoTpe

T.r T.F

f {wl *ta 3= <x'rQ>r$ 3=xr
,a1rcs'ua1nd "1"1;ui 1n:o1e:ado r nc up?ou pcpo

tsz) <x'rQyrQ=xr6'
:t0 ap
rlBq:rarualnbluTa€uuaoTTsaueuuT1rpx1rTmn?p,u1e{uendlas
ad gleuo6olro eTlJaToJd ap rolerado alsa
acnp axpo 1n:olerado pc
ur€n:esqo ES

(zzl qx'rg1rrf H=x I

: aTxss I
e5 Hax ?uauale un '{T0} Rlpulrouolro Ezeq o n3 uodEJ utr 'TTqp:Bdas
UaqTTH nlfeds un rryrapTsuog 'Tezeq TTlptlTeuobolro ezalodT uI earplpJl I
$€nla: EoXTlrrmolesOpz'ae?eeuuurnJoauroelft,oeTTaTzTetpq-raTpnTzseucoCu!
Tsu ap T.rnofTne?ccepTqen:sezlaTNrpulenXtT:olxdprFd sns ]
{
(Fpurouopo pzugl UagITH
rl.
nrfuds rn-Jlul l1uTbrpu TS TEIUTI rolpredo Fun saunTlcqU'U
{E
'Ep€Tp ap ln'lgdaycnuTonctopleiuteodpToAee u! aund alp, fmlsTTeulro] ulpcrEurau
iI
gTetlcads atequazasdat alSaunu es

T.r

(rzl ro> <r0rY 3=Y

:eereluszardag' (=pun[pe

fIvufi{s v?IIYrv tt tItflflllx 8-rT iltilsts I$ [it0cltc'[lYxtls

$!llt[[, cltcljltl $I $t$tBtx L4-n BtmBttl t[ aililzi sP[cTMr,t $BIIIAt

r. 0prnamnr Lanauprnerra$torrixTXonrcLrninrtrlar rp!EgsipAnTdrrtgHinuitoi gprer strpARABTLE 2.(

Foxtsa generald spafli Hilbert I

l- l..*, ...a, psgellxdreacborTleoaensete, nuadr infinit {
reprezentatE de o natrice cu un de linit I
ala cdrei elenante arn satlsfac condis,ia':
-_ :
V c*, k=1 ,2, {261
E[E 'r*o{ io,*i'<, Hilb
eler
tntr-adev6r, daci conslderdn un element aI unui spaliu Hilbert
&ecrls tn raport cu o bazi ortonornati Q* Ln
forna:

x=pOrc*; I't=/F[o*ft t,271

datorit[ liniarittrgii, actiunea operatorulul A se transfer& asupra

vectorllor bazei care devln f*:
=p trcr
(28)

Cum i.nsE t,..se pot expri-ura in raport cu aceeagi bazi g* ].n tonna:

S*-T 0ra* (2e)

rezult6 ci:

Ax=F p Ora*cr=F 0, ,p altc*) (30)

iffiT.-condl!,ia de nirglnlre a operatorului (l nxl <*) devine: (311
1"lt -'tr(-it

I

dHscnsoeloelnmlmnnbue&DsnelnttrairainigtutlenuilnnsliiloeeelutarptfleoasngrrTpuuai,ciebrnlpiaiurrJtoear.retn,ltll.ofloepimroernllriditce(riaexnlpga-orcil)esnuhpzilgilatepgiiTrnatlsxe:t(ar-iT,to,lleixncpsp-uue)tprelliraelina-l?rntxoirdop.npdrelteilecrrplaIacoetrefponateErtpnlrruatlupcuttocialirlanrcliztrd6enecae,xalcrlnoorpnrpdirnevmegerAstriaegnnpteooelalnrrltiitlnaarioll

spi.malrreetdlettlrauitclxPa.erae?roetldatccetue. lo)acprleiazdrasalrtnlefoiraclairpnpleleaenrsitpzmrauaaguteirlnli.cfelenmnrouletrnntpidarairnfdtelrnancstioetioren(sduapleneul.tdnaldurerue,nnhsoeiurrnnciiatlzecusert.e,t

: aTI35 as x TnTnluauBT€ Pf,dns€ v TnTnroxPredo EaunTlsp F; glTnZaU t
e
(eE) <n,Fe>_<x,r0> <rfi ,r0> 1.I t.F
[J-
f,=rrrre 3=td
t.
apun rT

(ss) ,*rn $_rr{,",o 3) Ht=, q,,"H,n t=nr*r b?'

: a.rpnr,r[?uo3 uI (I

(?s) qr4,,fqy=rfe rsgrfs H=t* (o

:sr.rf,s a?eod as g ln1n1feds pugul{-rede r[ e:ec1:o 'aX:ed PlTe ep ad (6
: Plf
(ss) ,*,, 3= (,0) s,r 3=('t', 3)"=* (ez
:u-aae rgq= tl r'urpq?oaulaEclueCauala
ed a111{ca1o:d gclglpout eE sxEc uf TnTnpou ea:ezlcerd ulrd D A;?
es elet uI TnTnpou earezlce:d u1:d
,Tezpq eToXueueTa EoTlTpou :Trnpou pnop u! aTrasep a?€od as XreqTTH (tz

n1{eds un-J?uI '11u16.r9u 1S reTuTf 'ts :o?e:ado Tntm eaunl{cq lfaq

ulte .,. Eue Tge ( ez)
t. *. ttuFl

(z€) [.*'"",.| - l/(1 _-lvl--l J vs lreql
lr*j,nrl= [x] tze Ezp rzp
IlVII
la*,tnt.| u.Ee il

zag trp

+i

: TPuoTsueur,rp xTUTJ
arlnonl?nppsornrsdpdoqo:duTnrEunoEX:A1eTurseaaddccon'nsETzads:aueuenox1Te1l3a6ecePr'elTl6rnne:Pllufcnocezraudr;n:sTdpn1'alnr:ureoocXlccHeauAaTepnleoelulueJe€uuTlerPaT3laas
e?EJapTsuoc Tt '?Tosd1[{t1tue1]6:gTue3JTorflTeJtnolaETrTaTduoTTTn'zl1€ecuuo1JsuarrutgupnulTeusT't]g
'rpTn3Tlred uI 'Z

tlvtntds vzIIru{ [t llllllllt OI-?I tnelst$ I$ [IIflcilc 'tlvttls

$[I1l!,X, CIICUItB $t SI$IBilB 14-tr ltBttfil Dx ilAilzt sPficTiltt $illIll

*Ar'=E ar,tr, gr) (g' x) (37) del

Cun, in cazul bazeiar ortonornate, produsul scalar a douX eLenente coin
x gl y din H se scrie: nota

(xty)=.8 or0r,F fr*rr= t38) 2,3
sep
E "rE F1(g1,or'=*E o;pro,r=* oip,=,o, P,
unr
Produsul scalar lntre x gi Ay este:
norl
T,<x, Ay) =E .iE ur:F.r=E E oiurrpr=F oiyr=.n, ( 3e ) Frtu

Foloslnd operatbrul adjunct A* avem; (40) cpe:
r.lrB,
(A+x, ,r=n* ul'rrpr=E E orrir) p,
91
91:
l.
(A*x,y>-E (arli) -9r. ( 41)

Constatin cE actlunea operatorului adjunct const6 din conjugarea

91 transpunerea natricel a' care il reprezint5.

ln generalrconjugarea unei expresii de forma <x,Ay) se reallzeazX

prln lnversarea ordhll elenentelor gl trecerea la adjunct pentru

op€rator:

<x, Ay>*=<yrA*x> .
si facem in continuare apel Ia conceptul de reductibilitate a
operatorllor. vom presupune ci operatorul A este slnplu gi nornal
(vectorll stri proprli genereaz6 tot spatlul H gl sunt ortogonall). h
ac€ste conditll, actiunea operatorului A asupra elenentelor x exprirnate
i.n raport cu vectorll sil proprii este
(- \-
ar,{ (0r) =E urtr0r=f l*erx=p FrO, (+zy
*=rlE urgrj=E
\l 'r

unde an tlnut seama de relatla ( 43 )
A[f),tft

Prln urmar€ B"= stf' 91 cele doui puncte de vederea prl:ul legat

reTlTpuoc ereJETxes lpr6eluT Tnalcnu erps u!

(6r) rp (r)x (r,? --, Q)I

))IJ =

:EU.:roJ qns Tg 'I

TiEluezgrder TJ Xd XpTur{cs-UsqTTH dTl ep TTrolprado rfltaufsn

t 'xpTuqrs-ueqlTH Tro?prado TtTunuep luns Ts

t8? ) trf

o)rr)zltvl 3

BTtETor 1""p *1i11ps TtTuTbrpu 11ro1ered6

gl't, *Elr*r,rlil=,r'or_d
l,*3 !,?
{rt} = 4rjrxF, rqry, nf I [, <*-v,nnl=tvl
rrertJyroJo $ --l

'reaepe-:1ug

tell =;rl

:rolPuun
e1l6fc1tm1rol 1uce! aTupTzJpeapzTellTelondeessRprupplou'Fc'up1lpnr1pn:ou1oen:ado
TnpoN u! pT139r€ ep
a1e llrdord eTTroT€A

(sr) {lxlxtlxd lx}tar=lvl

'nldlera ap'p11u13:ap r] eleod e'Jou plsperv 'Eu*ou o nc lroder * a"ilIli

reTuTT nTtpde ec lezlue6ro Tgnsu! Te TJ eleod lpp 1:eqfTH nTtrBdE un
ed TlTqIap rofTxoleredo :o:nlnl EarqiTnu 'lTluTup Tstr uP unc B9{

ITqsrpda$

ileqITH n1furls un ad lFIIep J0lured0 rnun su0il t',?

(rrl t0> <t0tY t=v

urlToI1e1l'lTs:napu-toFodlueETTezparperdulsaroar JesysrrX1pErnlz:ToeTlpepuro?ebTdepoTTpgp1Tcapcauru1onu6dsrrooele1E:larzdnQo-r)qulnspa'ePc.cerTppuue:TIop;s'pauT1eo{uerlT1ooeJup

vtvtlu{$ Iullftv t[ tlnffill zt_+lT Itltrsts Is tIInctIc'tltttl$

$tfftn, cItcul?[ $t $I$tHt[ 1-4-tu Btliltril ![ t[ttItA $PtcltAtA slfl

f-lxtt,r) ladc<* (so1 2,ll

Orlce operator Hllbert-schnldt poate fl exprlnat deci fie ca ReT
suni a unei eeril fle ca o integrald.
gel
2- funerrr or ux opERATon Fie A un operator simplu cu valorile net
proprll l" care formeazi {or}
el vectorll proprii baza (baza ia
reclproctr fllnd {0r}). Sunt valablte relalille:
e=$Llr=! tr6r> <0r; Pr.

o"i; ripr.? rior> <0, ( 51) ro|

<oiPrh definltle, o functle f de operatorul A, f(A) este operatorul: ^t
s1a) =D f (lr)0r) ( s2 )
in
pecaasrtreetlceauxnllaasrlt,litlicndlpLontter-zuancdiofm, epnlrulvclatireccaofnulnincelievadleovrlalerlapbroiliprciointprl"e.xdtn, re
re
a*=F tf$r> <0r,
5a
rrt=E errtQr)(0, (s3)
vl
RelTnen faptuL cE vectorii proprii ai unei fwctii de un optator
f(A) sunt aceiagi cu vectorli proprii ai operatorului inif,iaJ, in tinp
ce vaTorlle proprii sunt f(1,).
tn anumite conditli se poate vorbi gl de serll de operatori,
grobreme de convergentE etc.
l<1, De exemplu, daci norma operatorului A eete
etrict subunitarS, I lAl
are loc relalia:

F at -> (r-e) -1 (54)

ttU-0

deoarece dacE l lAl l=a cu a<1 gi definirn girul de operatori B-,

ao=Ea*, lr,l.Er (s5)

]_$ta"t<]gi u*=f+ (55)

uTxd EtrIrdxe e?Ed ea arEr (1)xI'=(?)A
ET TnT BrdnsE I rolBrado
"Ti3unl acnpuoo {X)x Tnun eaunl{cq
'(s'X)0=(s'?)0
'3ordT3o:olnp alse (s'l)o TnaTcnu prEp 're1ncl1red u;

(oe) uP(r)x1r't1-gi-1t1n

Ifi :Iat16E rp.rTo TaT4nqTrlsTp TnloqEfs atse 9 epun

{6s} to-s) g= (o'3 )s (t '8).0j

. eTleTar ecp1sTlps

eJer (s'?)0 eluopuedepuT rETrrTf aTeTEuuas uTp s?suroJ Tezsq ecordyca:
f '(1's)0 Eucq pugzTlTln euTuralap eleod as (s)n g1e:1ceds Bel€lTsueq
'TeaJ
pryrJ r 1n;r3aue:ad '(s)n eTTWt tJspuod pug^e sp(s'1)0 I.''!w lTufliuf
a;reluauaTl afvuwrs ep aySeuyqrcc o eJ Wrdxe as (X)x Tn1vuwes
r.lt lsp(sln rcfs XFuTluT eyTzepuod pug^c alTuqj (s'1)0 aJeluauraTa

€fpreps ap eyJeuyqrca o e3 gwJdxe as (1)x (eTpunl) Tnrerrues

! Trnpout

pnop uE ATlTnlq p?EX€rdfslul 1g aleod sne TEU ap gler6aluT pTn&rod

{8s) (r ) r0rr 3= ,, ,*

nc al6o1uue ulrd

'i
trs) Bp(8'3)0(e)nJ= (?)x

:e1er6e1u1 rofTrplTo^zap Eer€ldope ulrd 'p1ea: €xp pXpol ad g:o1ua e1

9JE3 nrXauErpd un gdnp alexepuT ',rer€lueureTa,r 11{cunl ep elT(l€rBurnuau

lTutJuT TnIt1nu Taun TTJFZTTTIn TnzEJ nrluad 'ezeezl1e:aua6

es 11{etmg aXTe op EI€TUTT oTtEuTquoc o Ec TTlounI Teun Ba.:eluaze:dag

alrJialuT eatrcnu

l'lep puo] qns rolTJolsJado Ig J0leFures PenrlJdxfl

''-(v-r)<-'g

:erTpp 'I<-(v-I)"g Fc glTnzer '-(--*-H T6 .*-E-I = "gq-"g um3

trtlrcxl$ YzrlvrY r{ !rtlffiIx rrT-rT nilsts ls tIi[0[It 'lTllrfs

$trilftlg, cncutTE sI SI$THI 14-tt lril1iltt Bx lfArrzA sr[cTflAr,l

,121 = [v(,e) 0 (t, sl ds= [ u(s) * ( r, s] ds { 61)

unde v(6) este densitatea spectrald a lui y in raport cu acelagi set
de functil de baz6riar f(t,s) reprezintX actiunea operatorului asupra
elemFeentdeeloar lbtai zpeia0rt(et,,se)L,emt(et,nste)=leTl0((tt,,ss;)
vectorii bazei O(t,s) in fotma: . pot exprjma in functie de

se

S (t, s) =ff to,s)0 ( t, o) ds (62)

r7tio, a) =/O lo, t)'{l {f , s)df t63)

1n aeeste condllii, v(tf "] scrle succesiv: {64}
y(il =rx(il = [u(s)S ( f,, s]ds=

/r/u(s) to, s)0( t,, oJ dsds=fv(o) S i c, s) do;

unde

,,tut =il, (o, s) u(s)ds {5s)

Sltuatia cea mai favorablli de actiune a operatorului T este aceea
i.n care vectorii bazel sunt vectori proprii ai operatonulul adic6

10(r, s) =l (s)0 (c, s) {56)

Este evldent cE v(s) =l (e) u(e) ( 67l

adLci densitatea spectrali a lui y(t), v(s) se obline prin {.nmulllrea
densitilil spectrale a lui x(t), u(s) cu l(s).

(er.r -1("""f1'-f*,+=tto--+t=|-(, lr#*ut3tYv--)tn(r)=Yr-=Ykt)-r'

)( sr rVy tx'/ff t=*r- (ry -y) = :r(rrr

)( t{. xra (y-rT) t="'" tV-*'a'V t=*,rr-",

(sr) 'Of *="'"| t=*'-o

tzLt ,Srnly t=*'alt t= (*),"

(rtl tSrDIy t=*taty *= {tStrty 3,o- ( (x)u).d=ey

(04) tr0rn (,?)J *=*tu i'?)'t t=:x (u)'i

11se:dxo ep :eTdu€xa ap TuaJgls.:Be1e1Auenle1c5urpy?ou'eTb:oallaer€ardaoTupsuugzou-Jl:'d+nugs
Jpumu eJps Tnr.m
Ber€rTJcs ellcred e1lqe:edas ?raqTTH tr1{eds ed 1{1u16:pu :o11:olerado
pBJezTfTln uf TrJdoJd roTT:oTEA TS rofTro?ceA ESTEZTTT?n
'ATl€rrJoJuT JPsPrr T3TU alPleJl
TJ ron nu
TJnzpr 1€ gleuoT{DunJ pzTTeuE ap e?€cTTap alcadse FJTIdU-p eleraue6
u! JoTTJo?EJ€do aTaJlceds ap a1e6e1 63T1€ural€ur alauelqo:d er

qzToerd'g TnTnso1eseclo Tntltads pzseuroJ TJoTeA Jolsaap ee?E1TIB1oJ,
'slenuT are nu (Iy-v) 1n:o1e:edo erec n:1uad I aITJoIpA luns :o1e:ado
Tnrm eTB TTrdoJd atT:oTpA Tceq 'sJanur aJe nu nllcadsa: Tnf,o?Eredo Tcap

(691 ,0=Jry*y*tTy (* 0*rS lO=r0(lrv-v) <= rory=rog

:g 1n1n.:olerado

aTp TT-rdord eTTroTp^ n:lued plsTxa Ru 'T erelduoc TSTTqeTTEA
sftroTEn pdnp F?sxapuT uTTTtreJ uTp TT.roleJsdo FO tlpAJasqo

(8el r (.rY--f )=lu
:TarlTffiJap uuoluoo 'V ?Pp :olerado
lnrm 1{efe1e tr:oleredo ap tV
aTTTllE] o 'a.TpnuTluoc u3 'm€lepTsuoC

IJotrsrado ap IIilmJ g'z

tIrut3t{5 vlIlYIV t{ ntffllI! gI.VT fltalsts rs ilt{lcilt 'tIvilil$

sHttA[[, c]tcutTt st st$IBilB L4-tt lt[tHtt tB iltlrzA sPIctRlil sEill tI

at1 -a41 1f , -lr;r-A1, -A1, =(1, -1.) A\At, (77 ) adici
pent
AA;A1A=A(A-II) -1= (A-lr+lr) (A-tr) -t=r+lit (78)
cl
Mal obgervSn cI <y,A^x>=g(l) este o funcSle de variabila conplexl l. \f(I

2.6 Scara spectrall valc

0 nultine de proiectorl P1 deplnzdnd de un paranetru real l, 1.
forneazi o scard spectrald daci sunt satisflcute conditiile:
a. a3B =) P-3Ps (P-P.=gr.p--t-'
b. P1 *o=P1 (p^ sunt continui Ia st6nga)
6. IimPl=O (P_-x-0, P_-=0) ,
l--- 0g)

O. l^inP1x=x (P,S<=x, P,--f)

Oaci e este un operator linlar autoadjunct de tip Hilbert-Schmidt
avAnd valorile spectrale (reale) I, ordonate crescdtor (vaLorile mi.nimf,
Si naxlnri fiind finite), sunele partlal"e:

F ^,.0*a0* , x) =E^x (80)

definesc o scari epectralX E^ care reprezintX prolectll pe un gir
ascenden, "te subspatli ortogonale. Si renarcim cX prolectorii slmpli
Pr nu defii.eec o scari spectrald.

in cazul- operatorllor Hilbert-schsridt spectrul este fomat dintr-un
set nunir:abil de valori conplexe avdnd suma pitratelor modulelor
flnlttr. DIn neferlclre, nulti operatorl uzuall nu sunt de tlp Hilbert-
Scbmidt. Generalizarea nodului de tratare al operatorilor de tip
Hilbert schnidt pe spalii Hilbert separabile Ia operatori necompactl
eate poslbili pentru operatorii autoadjuncti gi cei unitari,

Conceptul de scard spectralE se generalizeazi prin luarea in
csrglderare a unei fanlIll de operatorl indexati dupi valorll,e
variabllei reale I care genereazE o famlIle de operatorl de proieclie
P* analogtr celei prezentate in cazul operatorilor Hilbert-Schmidt.

Proprietetlle fan1l1el sunt identice ce cele prezentate anterior pentru

cazul nrmerabil, singura deosebire flind aceea cX I poate lua orice

valoare rea15.

Formal, generallzarea constE in inlocuirea sunelor cu lntegrale,
fanllla de operatori P, generAnd o misuri Lebesgue-sleltjes