Miimax i the theory of oerators o Hilbert saces ad Clarkso-McCarthy estimates for l q S ) saces of oerators i Schatte ideals Teresa Formisao A thesis submitted i fulfillmet of the Degree of Doctor of Philosohy of Lodo Metroolita Uiversity Lodo Metroolita Uiversity Jue 2014
Abstract The mai results i this thesis are the miimax theorems for oerators i Schatte ideals of comact oerators actig o searable Hilbert saces, geeralized Clarkso- McCarthy iequalities for vector l q -saces l q S ) of oerators from Schatte ideals S, iequalities for artitioed oerators ad for Cartesia decomositio of oerators All Clarkso-McCarthy tye iequalities are i fact some estimates o the orms of oerators actig o the saces l q S ) or from oe such sace ito aother
Cotets 1 Itroductio 3 I Miimax 8 2 Prelimiaries ad backgroud 9 3 Miimax ad semiorms 35 31 Itroductio 35 32 Miimax equality for semiorms 39 33 The miimax i reverse 42 34 A miimax theorem for oerators 45 35 Alicatio 48 36 Coclusio 49 4 Miimax ad Schatte ideals of comact oerators 51 41 Itroductio 51 42 Some miimax coditios for orms i S 57 43 Miimax coditio ad geometry of subsaces of Hilbert saces 64 44 Coclusio 75 1
II Estimates 77 5 Iclusios of saces l q S ) ad S H, K) 78 51 Backgroud 78 52 The saces B H, H ), S H, H ) ad l 2 S ) 86 53 Actio of oerators o l 2 S ) 94 54 The saces l q S ), l B H)) ad S H, K) 98 55 Iclusios of saces S H, H ) ad l S ) 112 56 Coclusio 119 6 Aalogues of Clarkso-McCarthy iequalities Partitioed oerators ad Cartesia decomositio 121 61 Backgroud o aalogues of McCarthy iequalities 121 62 Actio of oerators from B H ) o l q S ) saces 129 63 The mai result: The case of l q S ) saces 137 64 Uiform covexity of saces l S ) 141 65 Estimates for artitios of oerators from S 143 66 Cartesia decomositio of oerators 152 67 Coclusio 166 7 Coclusio 168 2
Chater 1 Itroductio The study of liear oerators ad fuctioals o Baach ad Hilbert saces aims at roducig results ad techiques that hel us to uderstad the structure ad roerties of these saces This study was develoed i twetieth-cetury ad attracted some of the greatest mathematicias such as D Hilbert, F Riesz, J vo Neuma ad S Baach It grew ad became a brach of mathematics called fuctioal aalysis It icludes the study of vector saces, saces of fuctios ad various classes of oerators defied o them Some of the most imortat theorems i fuctioal aalysis are: Hah-Baach theorem, uiform boudedess theorem, oe maig theorem ad the Riesz reresetatio theorem There are umerous alicatios of this theory i algebra, real ad comlex aalysis, umerical aalysis, calculus of variatios, theory of aroximatio, differetial equatios, reresetatio theory, hysics for examle boudary value roblems ad quatum mechaics), egieerig ad statistics Fuctioal aalysis uses laguage, cocets ad methods of logic, real ad comlex aalysis, algebra, toology ad geometry i the study of fuctios o liear saces ad fuctio saces The first miimax theorem was roved by vo Neuma i 1928 - it was a result related to his work o games of strategy No ew develomet occurred for the ext te years but, as time wet o, miimax theorems became a object of study ot oly i the game theory but also i other braches of mathematics Miimax theory 3
cosists of establishig suffi ciet ad ecessary coditios for the followig equality to hold: if su x X y Y fx, y) = su y Y if x X fx, y), 11) where fx, y) is a fuctio defied o the roduct of saces X ad Y Miimax theory is alied i decisio theory, game theory, otimizatio, comutatioal theories, hilosohy ad statistics, for examle to maximize otetial gai For overview o miimax theory ad its alicatios see [34] ad [15] This thesis has two aims ad, cosequetly, is divided ito two arts that corresod to them The first art cosists of Chaters 2, 3 ad 4 I these chaters we verify whether the geeral miimax coditios hold i various settigs of the oerator theory We also idetify ecessary ad suffi ciet coditios for which miimax theorems ca be roved for certai classes of fuctioals ad oerators o Hilbert saces The secod art cosists of Chaters 5 ad 6 Its aim is to obtai geeralized Clarkso-McCarthy iequalities for l q -saces of oerators from Schatte ideals S We aly these geeralized iequalities to rove various estimates for artitios ad Cartesia decomositio of oerators from S H, H ) ad l q S ) saces Boreshtei ad Shulma roved i [10] that if Y is a comact metric sace, X is a real iterval ad f is cotiuous, the 11) holds rovided that, for each y Y, the fuctio f, y) is covex ad, for each x X, every local maximum of the fuctio f x, ) is a global maximum Some weaker coditios o f that esure 4
the validity of 11) were established by Sait Raymod i [36] ad Ricceri i [33] Miimax theory has various alicatios i the oerator theory; see, for examle, Aslud-Ptak equality if su Ax λbx = su λ C x =1 x =1 if Ax λbx, where H is a Hilbert sace, x H, C is the the set of comlex umbers ad A ad B are bouded liear oerators o H [2] I our work we wated to idetify ew miimax theorems that hold for semiorms ad liear oerators that act o searable Hilbert saces I Chater 3 we obtai some miimax results that hold for sequeces of bouded semiorms We illustrate these results with examles of semiorms o the Hilbert sace l 2 Next we cosider ad rove some simle miimax formula for oerators The formula works also for biliear fuctioals o a Hilbert sace The mai results o miimax coditios obtaied i this thesis are the miimax coditios for oerators i Schatte ideals of comact oerators The details of this theory are exlaied i Chater 4, ad the results, amely Proositio 48 ad Theorems 49, 411 ad 415, have bee ublished i our joit aer i [19, 29-40] uder the joit authorshi of T Formisao ad E Kissi, where the secod author cotributed to various stages ad to its fial versio Clarkso roved i [12] famous iequalities for Baach saces of sequeces l, > 1 He used these iequalities to show that the l saces, for > 1, are ui- λ C 5
formly covex McCarthy obtaied i [28] o-commutative aalogues of Clarkso estimates for airs of oerators i Schatte ideals S Usig them, he roved that the saces S are uiformly covex, for 1 < <, ad therefore they are reflexive Baach saces [39, 23] The Clarkso-McCarthy estimates lay a imortat role i aalysis ad oerator theory They were geeralized to a wider class of orms that iclude the -orms by Bhatia ad Holbrook [6] ad Hirzallah ad Kittaeh [24] I [9] Bhatia ad Kittaeh roved aalogues of Clarkso-McCarthy iequalities for -tules of oerators of secial tye Kissi [25] exteded these estimates to all -tules of oerators He also exteded the results of Bhatia ad Kittaeh i [7] ad [8] o estimates for artitioed oerators ad for Cartesia decomositio of oerators I Chaters 5 ad 6 we develo a theory that allows us to exted the result of Kissi [25] ad to obtai a aalogue of geeralized Clarkso-McCarthy iequality for l q S ) saces We also establish various relevat results for oerators that belog to l q S ) ad S H, H ) saces Makig use of this, we rove that the saces l S ) are -uiformly covex, for 2 We also aalyze artitio of oerators from S saces ad Cartesia decomositio of oerators from l q S ) saces I fact, we exted the results obtaied i [25, Theorems 1 ad 4-5] to ifiite families of rojectios ad oerators This extesio requires ew techiques ad a ew aroach to the theory of l q S ) saces ad their relatio to S H, H ) saces Fially, we draw coclusios i Chater 7 We rovide elemetary backgroud 6
of the theory of Hilbert saces i the ext chater I most cases, the reader ca fid roofs of kow results i the refereced literature I some istaces, we give the roofs of some well kow results for the readers coveiece 7
Part I Miimax 8
Chater 2 Prelimiaries ad backgroud A liear sace X over R real umbers) or C comlex umbers) is called a ormed liear sace if it is equied with a orm, that is, each x X is associated with a o-egative umber x the orm of x, with the roerties: i) x 0 ad x = 0 if ad oly if x = 0; 21) ii) x + y x + y for all x, y X; 22) iii) αx = α x where α is a scalar 23) The distace betwee x, y X is defied by x y The cocet of orm geeralizes the otio of absolute value ad, more geerally, the otio of the legth of a vector For examle if R is the real lie with usual arithmetic ad x R the the usual absolute value, x, is a orm Havig the distace fuctio give by a orm, we ca exted familiar cocets from calculus to this more geeral settig Defiitio 21 Let x ) be a sequece i a ormed sace X, ) i) It is a Cauchy sequece if for every ε > 0 there is a iteger N such that m, N imlies x x m < ε ii) It has a limit x X i other words, x ) coverges to x) rovided that, for every ε > 0, there exists a iteger N such that N imlies x x < ε We write lim x x = 0, x x 0, lim x = x or x x A fuctio from X ito aother ormed sace Y is cotiuous at x X rovided 9
that for every sequece x ) i X covergig to x, the sequece f x )) coverges to f x) If every Cauchy sequece i a ormed liear sace X has a limit i X the X is said to be comlete A comlete ormed liear sace is called a Baach sace i hoour of the Polish mathematicia Stefa Baach Let X be a Baach sace with orm X = For 1 <, the l X) sace cosists of all ifiite sequeces x = x 1,, x, ) of elemets x X such that ) 1/ x = x < For =, the l X) sace cosists of all ifiite sequeces x = x 1,, x, ) of elemets x X such that x = su x < The roof below see Lemma 22 - Theorem 24) that all l X), 1 <, are Baach saces is based o roofs develoed i [30, 45-46] ad [32, 78-81] Recall that a real-valued fuctio f defied o a iterval I of R is covex if f αa + 1 α) b) αf a) + 1 α) f b), for all 0 α 1 ad all a, b I I other words, if a, b I the the grah of the fuctio f restricted to the iterval [a, b] lies beeath the lie segmet joiig the oits a, f a)) ad b, f b)) Positivity of the secod derivative is a suffi ciet coditio for covexity, showig that, i articular, the fuctio f t) = e t is covex 10
Cosider umbers 1, q 1 satisfyig 1 + 1 q If oe of the umbers is 1, we assume that the other is = 1 24) Lemma 22 [30, Lemma 236], [32, Lemma IX1] If s 0, t 0 the, for, q > 1 satisfyig 24), st s + tq q Proof If st = 0, the lemma is evidet Let s > 0 ad t > 0 Set a = l s ad b = q l t The s = e a/ ad t = e b/q Thus s = e a ad t q = e b By covexity of f t) = e t, we obtai st = e a/ e b/q = e 1 a+1 )b) 1 1 ea + 1 1 ) e b = 1 s + 1 q tq which comletes the roof The above lemma allows us to rove easily the followig imortat Holder s ad Mikowski s iequalities Proositio 23 i) [32, Theorem IX2] Holder s iequality) Let > 1, q > 1 satisfy 24) The, for ay N ad a i, b i C, i = 1,,, ) 1/ ) 1/q a i b i a i b i q 25) i=1 i=1 i=1 11
ii) [32, Theorem IX3] Mikowski s iequality) If 1 the, for ay N ad a i, b i C, i = 1,,, ) 1/ ) 1/ ) 1/ a i + b i a i + b i 26) i=1 i=1 Proof i) Let A = i=1 a i ) 1/ ad B = i=1 b i q ) 1/q If A = 0 or B = 0, the roof is evidet Otherwise, by Lemma 22, we obtai a i b i A B ai A ) + bi B q ) q i=1 = a i A + b i q qb q Thus a i b i AB A a i + AB qb q b i q Hece, summig u, we obtai i=1 a i b i AB A i=1 a i + AB qb q = AB i=1 1 + 1 ) q b i q = AB A A + AB qb q Bq = = AB ii) For = 1, the iequality is evidet Let > 1 ad let, q satisfy 1 + 1 q = 1 The 1 + q = ad alyig the triagle iequality, we have a i + b i = a i + b i a i + b i /q a i + b i ) a i + b i /q = = a i a i + b i /q + b i a i + b i /q Thus, summig u ad alyig 25), we get a i + b i i=1 a i a i + b i /q + i=1 b i a i + b i /q i=1 12
25) ) 1/ a i i=1 i=1 ) 1/q ) 1/ ) 1/q a i + b i + b i a i + b i i=1 i=1 If i=1 a i + b i = 0, the roof is evidet Otherwise, dividig the above iequality by i=1 a i + b i ) 1/q ad usig 1 = 1 1, we obtai q ) 1/ ) 1/ ) 1/ a i + b i a i + b i i=1 The roof is comlete i=1 i=1 Usig orm triagle iequality 22), we obtai that, for a Baach sace X with orm ad a i, b i X, i = 1,,, Mikowski s estimate gives ) 1/ a i + b i i=1 ) 1/ a i + b i ) 26) i=1 ) 1/ ) 1/ a i + b i i=1 i=1 27) We shall ow rove that all l X), 1 <, are Baach saces Theorem 24 Let X be a Baach sace The sace l X), for 1 <, is a Baach sace a ormed liear sace comlete with resect to Proof Let x = x 1,, x, ) l X) The x = 0 if ad oly if x = 0, ie, all x = 0 Clearly, αx l X) for each α C, ad αx = α x Let also y = y 1,, y, ) l It follows from Mikowski s iequality 27) that, 13
for all, ) 1/ x i + y i i=1 ) 1/ ) 1/ x i + y i i=1 i=1 ) 1/ ) 1/ x i + y i = x + y i=1 Hece the sequece of artial sums S = i=1 x i + y i is bouded by ad mootoe icreasig Therefore lim S exists ad Thus lim S = x i + y i i=1 i=1 x + y ) ) 1/ ) 1/ x + y = x i + y i = lim S x + y, i=1 so that x + y x + y Thus is a orm x + y ) The triagle iequality x + y x + y imlies that l X) is closed uder additio, ie, if x, y l X) the x + y l X) Thus l X) is a ormed liear sace ad we oly eed to show that it is comlete Let {x k = x k 1,, x k, )} k=1 be a Cauchy sequece i l X) The, for each ε > 0, there is N N such that, if r, s > N the ) 1/ x r x s = x r x s < ε Cosequetly, for each = 1, 2,, we have x r x s x r x s < ε Thus for each, the vertical sequece {x k } k=1 is a Cauchy sequece i X As X is a Baach sace ad, therefore, is comlete, there are x X such that lim k x k x = 0 for all 28) 14
Set x = x 1,, x, ) We shall show that x l X), ie, x < ad that x x k 0 As {x k } k=1 is a Cauchy sequece i l X), for ε = 1, choose N such that x k x N 1 for all k N Settig s = x N, we have x k = x k x N + x N x k x N + x N s + 1 for all k N Suose that x / l X) The there is q such that q x ) 1/ > s + 3 Hece, for all k N, q ) 1/ q ) 1/ s + 3 < x = x x k ) + x k 27) q ) 1/ q x x k + x k ) 1/ q ) 1/ x x k + q ) 1/ x k x x k + s + 1 By 28), we ca choose M N such that x x k 1, for each = 1,, q q 1/ ad all k M The q x x k q 1 q = 1 Combiig this with the above iequality, we have s + 3 < s + 2 This cotradictio shows that x l X) Let us show ow that x x k 0 As {x k } k=1 is a Cauchy sequece, choose N such that x k x N ε 9 for all k N 29) 15
For v 1, let Q v be the rojectio o l X) such that Q v y = y 1,, y v, 0, 0, ) for all y = y ) l X) The Q v y y ad Q v y y as v 210) For ε > 0, we ca choose m such that x Q m x < ε 3 ad x N Q m x N < ε 9 211) The, by 29)-211), we have for all k N, x k Q m x k x k x N + x N Q m x N + Q m x N x k ) 212) 2 x k x N + x N Q m x N < 2ε 9 + ε 9 = ε 3 By 28), we ca choose k 0 such that x x k all k k 0 The Q m x x k ) = m ) 1/ m x x k Hece, for k maxn, k 0 ), it follows from 211) - 213) that ε, for all = 1,, m ad 3m 1/ ) ε ) 1/ = ε 3m 1/ 3 213) x x k x Q m x + Q m x x k ) + Q m x k x k ε 3 + ε 3 + ε 3 = ε Thus x x k 0 as k Examle 25 [32, 78] Cosider the sace l = l C), for 1 < The elemets of l are sequeces of comlex umbers x = {x } 1 such that x < If we defie the -orm o l by the formula ) 1/ x = x, 16
the we derive from Theorem 24 that each l is a Baach sace Remark 26 Let X be a Baach sace The sace l X) is a Baach sace We omit the roof as it is similar to the roof of Theorem 24 It is also kow see for examle [16, Lemma 9, XI9]) that l l q ad x x q, for x l, if 1 q Thus l 1 is the smallest ad l is the largest of the saces Defiitio 27 [1, 2] Let X be a liear comlex sace A ier-roduct or scalar-roduct), ) is a comlex-valued fuctio defied o X X which satisfies the coditios: 1 x, y) = y, x); 2 αx + βy, z) = α x, z) + β y, z) for α, β C 3 x, x) 0, with equality if ad oly if x = 0 We ca derive from the above coditios that x, αy + βz) = αy + βz, x) = α y, x) + β z, x) = α y, x) + β z, x) = α x, y) + β x, z) The Cauchy-Schwarz-Buyakovsky iequality is oe of the most imortat iequalities i mathematics: 17
Theorem 28 Cauchy-Schwarz-Buyakovsky iequality) [1, 2] If, ) is a ierroduct o a liear sace X the x, y) x, x) 1 2 y, y) 1 2, for all x, y X, 214) with equality if ad oly if x ad y are liearly deedet Proof If x, y) = 0 the theorem is roved We ca assume that x, y) 0 Lettig θ = x,y), we fid from Defiitio 27 that, for ay real λ, x,y) 0 θx + λy, θx + λy ) = θ 2 x, x) + λθ x, y) + λθx, y) + λ 2 y, y) 2 = x, y) x, y) x, y) x, y) x, x) + λ x, y) + λ x, y) x, y) x, y) + λ2 y, y) x, y) 2 y) 2 = x, x) + λ + λ x, x, y) x, y) + λ2 y, y) = λ 2 y, y) + 2λ x, y) + x, x) 215) We arrived at a o-egative o roots or oe reeated root) quadratic i λ Thus the discrimiat of this quadratic is o-ositive: 4 x, y) 2 4 y, y) x, x) 0 Hece x, y) 2 x, x) y, y) ad we have the iequality 214) The equality i 214) holds if ad oly if the quadratic has a reeated root, i other words if ad oly if λ 2 y, y) + 2λ x, y) + x, x) = 0, for some λ R 18
This imlies see 215)) that θx + λy, θx + λy ) = 0 Thus θx + λy = 0 for some real λ, so that the vectors x ad y are liearly deedet Let X be a liear sace with scalar roduct, ) Set x = x, x) 1/2 Let us check that is a orm o X From the Defiitio 27 we have x 0 with equality if ad oly if x = 0 Additioally, it follows that αx 2 = αx, αx) = αα x, x) = α 2 x 2 for all scalars α Thus αx = α x for all scalars α To rove the triagle iequality, we aly the Cauchy-Schwarz-Buyakovsky iequality to obtai x + y 2 = x + y, x + y) = x, x) + x, y) + y, x) + y, y) 214) x 2 + 2 x y + y 2 = x + y ) 2 for all x, y X This imlies the triagle iequality x + y x + y for all x, y X A Baach sace whose orm comes from a scalar-roduct as x = x, x) 1 2 is called a Hilbert sace i hoour of the Germa mathematicia David Hilbert [32] A ormed liear sace ot comlete) is called a re-hilbert sace if its orm comes from a ier-roduct Hilbert ad re-hilbert saces are called ierroduct saces [32] 19
Examle 29 [1, 5-7] Cosider the Hilbert sace l 2 that cosists of sequeces x = {x } 1 of comlex umbers such that x 2 < As i Examle 25, it is a Baach sace with orm ) 1/2 x = x 2 The scalar roduct i the sace l 2 has the form x, y) = x y The series o the right coverges absolutely because ts = ts 1 2 t 2 + 1 2 s 2 for all t, s C We omit the simle rove that the umber x, y) satisfies all the coditios of a scalar roduct ad the orm x of each vector x l 2 satisfies x = x, x) 1/2 = x 2 ) 1 2 Defiitio 210 [32] [27, Defiitio 121] Let X be a ier-roduct sace Elemets x, y X are orthogoal we write x y) if their ier-roduct x, y) = 0 For sets A ad B i X, we write A B if x, y) = 0 for all x A ad y B Fially, A is the set of all vectors x X such that x y for all y A; for ay set A this is always a subsace of X, moreover sice A = a A {a}, A is a closed subsace by cotiuity of the ier roduct 20
A subset S of X is a orthogoal set, if x, y S ad x y imly x, y) = 0 If each elemet of a orthogoal set S has orm 1, the S is a orthoormal set A orthoormal set S i X is comlete if S T ad T is aother orthoormal set i X imly S = T Oe of the most used results i all mathematics ad esecially i fuctioal aalysis is a result take from logic ad it s called Zor s lemma It was stated without roof by the ma whose ame it carries [32] I fact it is ot ossible to rove Zor s lemma i the usual sese of the world However,it ca be show that Zor s lemma is logically equivalet to the axiom of choice, which states the followig: give ay class of o-emty sets, a set ca be formed which cotais recisely oe elemet take from each set i the give class The axiom of choice is ituitively obvious We therefore treat Zor s lemma as a axiom of logic [38] Other, equivalet forms of Zor s lemma iclude: Pricile of choice, Pricile of trasfiite iductio, Zermelo theorem Every set ca be well ordered), the Tukey- Teichmuller theorem ad Hausdorff s theorem Zor s lemma is frequetly used i lace of trasfiite iductio, sice it does ot require the sets cosidered to be well ordered Usually sets are aturally equied with a artially ordered relatio but ot ecessary a well ordered relatio [31] Defiitio 211 [32] [31] Let P be a set ad R a relatio o P satisfyig for x, y, z P the followig three coditios: 21
1 reflexive) xrx 2 atisymmetric) xry, yrx imlies x = y 3 trasitive) xry, yrz imlies xrz The P, R) is a artially ordered set If additioally, every two elemets of P are comarable ie for x, y P either xry or yrx, the the set P is totally ordered or liearly ordered) If S P the m P is a uer boud for S if srm for all s S ad a lower boud if mrs for all s S A smallest largest) elemet i S is a elemet s S which serves as a lower boud uer boud) for S) A well-ordered set is a artially ordered set every o-emty subset of which ossesses a smallest elemet A elemet m P is maximal rovided a P ad mra imlies m = a Lemma 212 Zor s lemma)[32] [38] [31]Let P be a artially ordered set ad suose every totally ordered subset S has a uer boud i P The P has at least oe maximal elemet Theorem 213 [32] Let X {0} be a ier-roduct sace The X cotais a comlete orthoormal set Proof Proof of this theorem uses Zor s lemma Let x 0 be i X The { } s = is a orthoormal set Let P be the collectio of all orthoormal sets x x 22
cotaiig s ad ordered by iclusio Let P 0 be ay liearly ordered totally ordered) subset of P Cosider S 0 = UεP 0 U Let x, y S 0 The x U 1 ad y U 2 Sice P 0 is liearly ordered we ca assume that U 1 U 2 Thus x, y U 2 Sice all elemets of P 0 are orthoormal sets we have that x y ad so S 0 is a orthoormal set Thus S 0 P S 0 is clearly a uer boud for P 0 sice for every U P 0, we have U S 0 By Zor s lemma, P has a maximal elemet T Suose that T is ot a comlete orthoormal set i X The there exists a elemet z X such that z / T ad T {z} is a orthoormal set This imlies that T is ot a maximal elemet i P ad we have a cotradictio Thus T is a comlete orthoormal set i X ad the theorem is roved Theorem 214 [32, 20] Let H be a Hilbert sace ad S a comlete orthoormal set i H The x = x, u) u, for every x H, u S where the covergece is ucoditioal the series coverges to the same elemet if we rearrage the elemets of the series), the umber of u S, for which x, u) 0, is at most coutable ad x 2 = u S x, u) 2 the Parseval equality) 216) 23
If H is searable ie it cotais a coutable dese subset), the ay comlete orthoormal set S is coutable, say S = {u }, ad x = x, u ) u ad x 2 = x, u ) 2 A Hilbert sace H is the direct sum of its closed subsaces M ad N, ie M N = H if M N = {0} ad each z H ca be writte i the form z = x + y, where x M ad y N As M N = {0}, this reresetatio of z is uique Theorem 215 [17, Theorem 224] For every closed subsace L of a Hilbert sace H, L L = H I this thesis we study searable Hilbert saces Defiitio 216 [27, 31] Let X ad Y be ormed liear saces A ma T : X Y is a liear trasformatio, liear oerator or oerator i this thesis all oerators are liear) if T αx + βy) = αt x + βt y, for all x, y X ad α, β C It is bouded if there exists M 0 such that T x M x for all x X The orm T of a bouded oerator T ca be defied as T = su x 1 T x, or equivaletly T = su x =1 T x = su x X T x x 24
If T is bouded, oe-to-oe, oto ad its iverse T 1 is bouded, the T is a isomorhism ad we say that the saces X ad Y are isomorhic Theorem 217 [27, 32] The collectio B X, Y ) of all bouded oerators from a ormed liear sace X to a ormed liear sace Y is a ormed liear sace i the oerator orm, where the vector oeratios are defied oitwise If, i additio, Y is a Baach sace, the B X, Y ) is a Baach sace Whe X = Y we deote B X, Y ) as B X) Theorem 218 [38, 219-220] Let X, Y be ormed saces ad let T : X Y be a oerator The followig are equivalet: 1 T is bouded; 2 T is cotiuous at 0; 3 T is cotiuous o all of X Examle 219 [27, Examle 28] Let H be a Hilbert sace with orthoormal basis {e } ad {α } a bouded sequece of comlex umbers Set Ae = α e Exted A by liearity ad cotiuity to all of H The, give x H, we have x = x, e ) e ad Ax 2 = Ax, e ) 2 = su x, e ) 2 α 2 ) α 2 x, e ) 2 = 2 su α ) x 2 25
We see that A is bouded ad A su α Cosideratio of Ae shows that A = su α Such a oerator A is called a diagoal oerator, with diagoal sequece {α } Defiitio 220 [30, 86] Let X be a Baach sace ad let X deote the liear sace of all bouded liear oerators from X ito C Every f X is called a liear fuctioal ad f = su { f x) : x 1} is its orm The sace X is the dual or cojugate) sace of X Theorem 221 [27, 36] adjoit of a oerator) Give Hilbert saces H ad K ad T B H, K), there is a uique T B K, H) such that T x, y) K = x, T y) H for all x H ad y K The oerator T is called the adjoit of T ad see [42, age 78]) T = T Defiitio 222 [30, 93] A oerator T B H) is self-adjoit if T = T Theorem 223 [30, 93] A bouded oerator T is self-adjoit if ad oly if T x, x) is real for all x H 26
With every oerator T : X Y we associate two imortat subsaces: The ull sace or the kerel deoted by ker T ) ad the rage or the image of T deoted by R T ) The ull sace cosists of all x X such that T x = 0 ad the the rage cosists of all y Y such that T x = y for some x X The subsace RT ) is ot ecessarily closed i Y, while ker T is always a closed subsace of X [26, age 52] Theorem 224 [30, Proositio 427] For all T B H) : a) ker T ) = R T ) ; b) ker T ) = R T ) Let A be a bouded liear oerator o a Hilbert sace H The orm of A see Defiitio 216) is } A = su { Ax : x = 1} = su {Ax, Ax) 1 2 : x = 1 From Cauchy-Schwarz-Buyakovsky iequality we obtai for all x H su Ax, y) su Ax y ) = su Ax = Ax y =1 y =1 y =1 O the other had, let x = 1 ad Ax 0 Set y 0 = Ax Ax The y 0 = 1 ad Ax = Ax, Ax) Ax = Ax, y 0 ) su Ax, y) y =1 Hece su Ax, y) = Ax 217) y =1 27
Thus A = su Ax = su Ax, y) x =1 x = y =1 Defiitio 225 [30, 93] A oerator T B H) is ositive if T x, x) 0 for all x H It is clear that 0 ad 1 are ositive, as are T T ad T T for ay oerator T B H), sice for all x H, we have T T x, x) = T x, T x) 0 ad T T x, x) = T x, T x) 0 For oerators A ad B, A B is defied to mea that A B 0; equivaletly A B Ax, x) Bx, x) for all x Theorem 226 [30, Theorem 432] Give ay ositive oerator T, there is a uique ositive oerator A such that A 2 = T The oerator A is deoted by T 1/2 Moreover, T 1/2 commutes with ay oerator that commutes with T Defiitio 227 [30, 95] If H ad K are Hilbert saces ad a oerator U B H, K), the U is uitary if U U = 1 H ad UU = 1 K Defiitio 228 [38, 237] A rojectio P o a Baach sace B is a idemotet i the algebra of all liear bouded oerators o B, that is, P is a liear bouded trasformatio of B ito itself such that P 2 = P Projectios ca be described geometrically as follows [38, 237] here the symbol reresets direct sum of subsaces): 28
1 If P is a rojectio o a Baach sace B, the the rage RP ) is a closed subsace of B ad B = RP ) kerp ); 2 a air of closed liear subsaces M ad N of a Baach sace B, such that B = M N determies a rojectio P whose rage ad ull sace are M ad N, resectively If z = x + y is the uique reresetatio of a vector i B as a sum of vectors i M ad N, the P is defied by P z = x) I the theory of Hilbert saces we cosider rojectios, sometimes called orthogoal rojectios, whose rage ad ull sace are eredicular, ie, ker P = R P )) Defiitio 229 [30, 94] A oerator P B H) o a Hilbert sace H is a orthogoal rojectio, or ortho-rojectio, if P = P ad P 2 = P We will call such oerator P just rojectio By the rojectio theorem see [27, 13]), every o-zero orthogoal rojectio is of orm 1 We say that two rojectios P ad Q are orthogoal if P Q = 0 It ca be roved [38, 275] that P Q = 0 QP = 0 R P ) R Q) The followig defiitio holds for Baach saces but we shall oly cosider Hilbert saces 29
Defiitio 230 [17, 59] i) A set K i a Baach sace X is called a recomact set if, for every sequece {x } i K, there exists a elemet x X a limit oit) ad a subsequece {x i } of {x } such that x i x It is comact, if all limit oits also belog to K ii) A liear oerator A : X Y, where X ad Y are Baach saces, is called a comact oerator if ad oly if, for every bouded sequece {x } i X, the sequece {Ax } is a recomact set Clearly, a comact oerator must be bouded, sice the image of the uit ball of X must be a bouded set i Y otherwise, we ca easily fid a sequece {x } iside the uit ball of X such that Ax ad, therefore the set {Ax } has o covergig subsequece) [17, 59] Theorem 231 [43, 10] If T is a comact oerator o a Hilbert sace H, the for ay bouded liear oerator S o H, the oerators T S ad ST are both comact If S is also comact, the T + S is comact Note that if T is comact, the αt = α1) T is also comact for all comlex umbers α Theorem 232 [43, 11] A bouded liear oerator T o H is comact if ad oly if T is comact, if ad oly if T T is comact, if ad oly if T = T T ) 1/2 is comact 30
Theorem 233 [43, 11] If {T } is a sequece of comact oerators o H ad T T 0 for some oerator T o H, the T is also comact Defiitio 234 [30, 168] The sectrum of a oerator T B H), deoted by σ T ), is the set of all scalars λ such that T λ1 is ot ivertible i B H) Theorem 235 Sectral Theorem for Comact Oerators) [30, Theorem 916 ad 918] Let T be a comact oerator i B H) i) The set σ T ) = {λ } is fiite or coutable All λ 0 are eigevalues ad the corresodig eigesaces M are fiite-dimesioal If {λ } is coutable ifiite the λ 0, as ii) If T is self-adjoit, the all λ are real, all eigesaces M are mutually orthogoal ad their closed liear sa is all of H Moreover, T = λ P, where P are rojectios o M We will eed the followig versio of the sectral theorem also called the Schmidt reresetatio see [32, 64, 75]) Corollary 236 [27, Corollary 425] Let T be a comact self-adjoit oerator o a searable Hilbert sace H The there is a orthoormal basis {e } of H cosistig of eigevectors for T such that T x = λ x, e ) e, for each x H, 31
where λ is the eigevalue of T corresodig to the eigevector e I [21] the authors aalyze comletely cotiuous oerators that ma weakly coverget sequeces to orm coverget sequeces I our case of oerators o searable Hilbert saces, comletely cotiuous oerators coicide with comact oerators, sice, for reflexive saces, the two defiitios are equivalet we kow that all Hilbert saces are reflexive, ie, if H is a Hilbert sace the it is isomorhic to its secod dual H ) [11] [43] [38] Defiitio 237 [38, 208] A algebra real or comlex) is a liear sace A equied with a multilicatio oeratio that assigs to each x, y A a elemet xy A such that, for all x, y, z A ad scalars α, the followig axioms must be satisfied: 1) Associative law) x yz) = xy) z; 2) Distributive laws) x y + z) = xy + xz ad x + y) z = xz + yz; 3) Law coectig multilicatio ad scalar multilicatio) α xy) = αx) y = x αy) A algebra is commutative if xy = yx for all elemets of the sace Defiitio 238 [38, 302] A Baach algebra is a real or comlex Baach sace B, which is also a algebra i which the multilicative structure is related to the orm by the followig requiremet xy x y for all x, y B 32
For examle, the liear sace BH) of all bouded oerators o a Hilbert sace H edowed with the oerator orm is a Baach algebra, where multilicatio of oerators is their comositio Defiitio 239 [38, 324] A Baach algebra A is called a Baach -algebra if it has a ivolutio, that is, if there exists a maig x x of A ito itself with the followig roerties: 1) x + y) = x + y for x, y A; 2) αx) = αx for x A ad α C; 3) xy) = y x for x, y A; 4) x = x for x A; 5) x = x for x A If H is a Hilbert sace, the the algebra BH) of all bouded liear oerators o H is a Baach -algebra with the adjoit oeratio T T as the ivolutio A subalgebra of the algebra BH) is said to be self-adjoit, or a -subalgebra, if it cotais the adjoit of each of its oerators All closed self-adjoit subalgebras of B H) are Baach -algebras Moreover, the closed self-adjoit subalgebras of B H) that satisfy the followig coditio: xx = x 2, for all elemets x, costitute a secial class of Baach -algebras called C -algebras 33
Defiitio 240 [38, 209] Let A be a comlex algebra Its subset I is a left resectively, right) ideal of A, if 1) αa + βb I for all a, b I ad α, β C; 2) ab I resectively, ba I) for each a A ad b I It is a two-sided ideal of A, if it is a left ad a right ideal of A Let C H) deote the set of all comact oerators o H From the above Theorems 231, 233 ad 232 we kow that C H) is a closed, self-adjoit subalgebra ad a two-sided ideal of the algebra B H) Thus C H) is a C -subalgebra of B H) It is kow that C H) is the oly roer closed two-sided ideal of B H) [43, 12] 34
Chater 3 Miimax ad semiorms 31 Itroductio Let X ad Λ be sets ad let f be a real fuctio o X Λ = {x, λ) : x X, λ Λ} Recall that the miimax equality is the followig equality: ) if su f x, λ) λ Λ x X ) = su if f x, λ) x X λ Λ As we shall see i Proositio 31, the iequality ) if su f x, λ) λ Λ x X ) su if f x, λ) x X λ Λ holds for all fuctios f Therefore to rove the miimax equality, oe oly eed to rove the iverse iequality ) if su f x, λ) λ Λ x X ) su if f x, λ) x X λ Λ We give below the roof of the followig kow roositio, as we could ot fid a referece Proositio 31 Let X ad Λ be sets ad let f be a fuctio from X Λ = {x, λ) : x X, λ Λ} ito R The ) if su f x, λ) λ Λ x X ) su if f x, λ) x X λ Λ 35
Proof For every µ Λ, we have su x X f x, µ) su if f x, λ) x X λ Λ Thus This cocludes the roof if su µ Λ x X f x, µ) su if f x, λ) x X λ Λ To rove some theorems for examle Theorem 411 we eed the followig lemma The lemma is kow, but we could ot fid ay referece Lemma 32 Let f : X Λ R be a fuctio o the roduct of o-emty sets X ad Λ Suose that there exists µ Λ such that su f x, λ) = f x, µ) for each x X 31) λ Λ Alteratively, suose that there exists x 0 X such that if f x, λ) = f x 0, λ) for each λ Λ 32) x X The ) if su f x, λ) x X λ Λ ) = su if f x, λ) = if f x, µ) 33) λ Λ x X x X Proof Alyig Proositio 31, we always have ) if su f x, λ) x X λ Λ ) su if f x, λ) 34) λ Λ x X 36
Suose ow that 31) holds The ) if su f x, λ) x X λ Λ = if f x, µ) x X Hece ) if su f x, λ) x X λ Λ ) = if f x, µ) su if f x, λ) x X λ Λ x X Combiig this with 34), we obtai 33) The roof that 32) imlies 33) is similar I sectios 32 ad 33 we cosider the validity of the miimax equality for a sequece of semiorms o Baach saces Defiitio 33 [35, 12] Let X be a comlex vector sace A o-egative, fiite, real-valued fuctio g o X is called a semiorm if, for all x, y X ad scalars λ, g λx) = λ g x) 35) g x + y) g x) + g y) 36) I fact, ay fuctio o X that satisfies 35) ad 36) is o-egative Ideed, for each x X, g 0) = g 0x) = 0 g x) = 0, so that 0 = g 0) = g x + x)) g x) + g x) = g x) + 1 g x) = 2gx) Clearly, the set g 1 0) is a liear subsace of X If g x) = 0 imlies x = 0, the see21)-23)) g is a orm, so that X, g) is a ormed sace 37
Defiitio 34 [1, 36] A semiorm g o a ormed liear sace X is bouded if there exists M > 0 such that g x) M x for all x X For examle, let X be a Baach sace with orm For each bouded oerator T o X, we have that g T x) = T x, for x X, is a bouded semiorm o X, as g T λx) = T λx = λ T x = λ g T x); g T x + y) = T x + y) T x + T y = g T x) + g T y) ad g T x) = T x T x for all x X A bouded semiorm g o X defies a equivalet orm o X ad we will write g, if there exists 0 < k such that k x gx) for all x X I other words, g if gx) k = if x X x = if gx) > 0 37) x =1 For examle, if T is a bouded oerator o X that has bouded iverse T 1 the g T, as x = T 1 T x T 1 T x = T 1 g T x) for all x X, so that T 1 1 x g T x) ad k = T 1 1 38
It follows from 37) that g if ad oly if there is a sequece {x } i X such that x = 1 for all ad gx ) 0, as The followig theorem about semiorms is kow Note that if the semiorms are liear, the the roof of the theorem follows from the uiform boudedess ricile ad the Baach-Steihaus theorem see for examle [27, Theorems 311 ad 312]) Theorem 35 [1, 37] Let {g k } k=1 be a sequece of bouded semiorms o a Hilbert sace H If the sequece is bouded at each oit x H, the the fuctio defied by is also a bouded semiorm g x) = su g x) for x H, 32 Miimax equality for semiorms Let {g k } k=1 be a sequece of bouded semiorms o a Hilbert sace H bouded at each oit x H Cosider the miimax formula: if su g x) = su if g x) x =1 x =1 By Theorem 35, g x) = su g x) is a bouded semiorm o H Hece the miimax formula takes the form if gx) = if x =1 x =1 su g x) = su if x =1 g x) 38) 39
A articular case could be that there exists m N such that g m x) = su g x) Proositio 36 Let x = x, x) 1/2 be the orm o a Hilbert sace H Let {g k } k=1 be a sequece of bouded semiorms o H bouded at each oit x H ad let g x) = su g x) The i) ii) iii) If g the 38) holds If g but all g the 38) does t hold Let g The 38) holds if ad oly if for each ε > 0 there exists ε such that g ε ad if x =1 g ε x) if x =1 gx) ε Proof i) We kow that g if ad oly if there is a sequece {x } i X such that x = 1 for all ad gx ) 0, as The 0 if gx) if gx) = 0 x =1 x {x } Thus if x =1 gx) = 0 We kow from Proositio 31 that if gx) = if x =1 su x =1 g x) su if g x) x =1 As all semiorms are o-egative, we have su if x =1 g x) 0 Thus if gx) = if x =1 su x =1 g x) = su if g x) = 0 x =1 ii) Suose that g but all g Thus for each there exists sequece {x j } j=1 such that x j = 1, for all, j, ad, for each, gx j ) 0, as j Hece for all we have if g x) if g x) = 0 x =1 x {x j } j=1 40
Therefore su if x =1 g x) = 0 O the other had, g ad by 37) if gx) > 0 Thus x =1 if gx) = if x =1 su x =1 g x) > 0 Therefore LHS > 0 ad RHS = 0 Hece the miimax 38) does t hold iii) Let k = if x =1 g x) ad k = if x =1 gx) The k > 0 if ad oly if g The miimax 38) holds if ad oly if k = su k, that is, for each ε > 0 there exists ε such that g ε ad if x =1 g ε x) if x =1 gx) ε Case ii) is a subcase of iii) but we thik that it is worth metioig it as idividual case for clarity Examle 37 below illustrates Proositio 36 case ii) whe g = 2 Examle 37 Cosider the Hilbert sace ) 1/2 l 2 = x = {x } 1 : all x C, x 2 = x 2 < ad the followig semiorms g o l 2 give by g x) = i=1 x i 2) 1 2 where x l 2 The roof of coditio 35) is obvious ad the roof of the triagle iequality called i this case the Mikowski s iequality ) 1 2 g x + y) = x i + y i 2 i=1 i=1 ) 1 2 x i 2 + i=1 y i 2 ) 1 2 was obtaied i Proositio 23 Thus g are semiorms o l 2 = gx) + gy) We have that gx) = su g x) = su i=1 x i 2 ) 1 2 = i=1 x i 2 ) 1 2 = x 2 41
Thus LHS = if gx) = if x 2 = 1 x 2 =1 x 2 =1 O the other had RHS = su Thus the miimax formula does ot hold if g x) = su 0 = 0 x 2 =1 33 The miimax i reverse Let H be a Hilbert sace Let {g k } k=1, be a sequece of semiorms i H such that g m x) = if g x) for all x H ad some m N for examle {g k } k=1, could be mootoe icreasig, ie g k x) g k+1 x) for all x H ad we ca set m = 1) Cosider the miimax formula, which is the reverse to miimax 38) if Theorem 38 The miimax formula 39) holds ad su g x) = su if g x) 39) x =1 x =1 if Proof The iequality i the formula su g x) = su if g x) = su g m x) 310) x =1 x =1 x =1 if su g x) su g m x) = su if g x) x =1 x =1 x =1 is obvious, as the ifimum over of su x =1 g x) is ot greater tha su x =1 g m x) The reversed iequality holds for all miimax formula see Proositio 31) Hece Eq 310) holds 42
Examle 39 Let us cosider the followig three examles i) The followig semiorms g o l 2 g x) = i=1 x i 2 ) 1 2 where x l 2 are mootoe icreasig ad RHS = su if g x) = su if x =1 x =1 LHS = if su g x) = if x =1 su x =1 i=1 i=1 Thus the reversed miimax 310) holds as equality x i 2 ) 1 2 x i 2 ) 1 2 = su x 1 = 1 x =1 = if 1 = 1 ii) Cosider the followig semiorms g o l 2 ) 1/ g x) = P x = x i i=1 for x l 2 We have g 1 x) = x 1 g x) for all N ad all x l 2 Thus g 1 x) = if g x) for all x l 2, ) 1/ ) 1/ ) 1/2 x i x i x i 2 = 1, i=1 i=1 i=1 for x 2 = x = 1, ad all N, 2 Hece su g x) 1 for all N ad if su g x) if 1 = 1 x =1 x =1 43
We kow that if g x) = g 1 x) = x 1, for all x l 2 Thus su if g x) = su x 1 ) = 1 x =1 x =1 Hece, it follows from Proositio 31 that 1 if su g x) su if g x) = su g 1 = 1 x =1 x =1 x =1 Thus the reversed miimax 310) holds as equality iii) Let us cosider the followig semiorms S o l 2 S x) = P x 1+ 1 = x i 1+ 1 i=1 ) 1/1+ 1 ) for x l 2 From Mikowski s iequality we kow that S s are semiorms As the fuctio f t) = i=1 st i) 1/t, 0 < t, s j > 0) is oicreasig [21, 92], we obtai that the sequece {S x)} is mootoe icreasig, ie 1 + 1 > 1 + 1 + 1 imlies x 1+ 1 +1 x 1+ 1 We have S x) max i=1,, x i ) x 1, for all 1 N Thus if S x) x 1 O the other had, if S x) S 1 x) = 1 x i 1+ 1 1 i=1 Hece if S x) = S 1 x) = x 1 Therefore we have ) 1/1+ 1 1) = x 1 RHS = su if S x) = su x 1 = 1 x =1 x =1 44
Let us ow calculate LHS We have that, for all, su S x) S 1, 0,, 0, )) = 1 x =1 Thus LHS = if su x =1 S x) if 1 = 1 O the other had LHS = if su S x) su S 1 x) = su x =1 x =1 x =1 1 Thus the reversed miimax 310) holds as equality i=1 ) 1/2 x i 2 = su x 1 = 1 x =1 34 A miimax theorem for oerators Let H be a Hilbert sace ad let A be a bouded liear oerator o H The uiform orm of A see Defiitio 216) is } A = su { Ax : x = 1} = su {Ax, Ax) 1 2 : x = 1 where x, y) is the scalar roduct of elemets x, y H Defiitio 310 [30, 63] A liear oerator T B H) is bouded from below if there is a k > 0 such that, T x k for all x H, x = 1 Clearly, beig bouded below imlies that T is ijective as ker T ) = {0} However, the coverse is ot true i ifiite-dimesioal saces Theorem 311 The bouded iverse theorem) [30, Theorem 36] For a ijective liear oerator T B H), the followig are equivalet: i) T 1 is bouded; 45
ii) T is bouded below; iii) R T ) is closed I this thesis reort we will cosider various cases whe miimax formula holds withi the theory of oerators o Hilbert saces We will start with the followig simle case of miimax formula Theorem 312 Let H be a Hilbert sace ad A a bouded oerator o H i) If A is ivertible ad a) dim H = 1, the the miimax formula: if x =1 { su Ax, y) y =1 } { } = su if Ax, y) y =1 x =1 = a 311) holds, where a is a scalar such that Ax = ax for all x H b) dim H > 1, the the miimax coditio does ot hold: { } if x =1 su Ax, y) y =1 = if Ax = k > 0, 312) x =1 while { } su y =1 if x =1 Ax, y) = 0 313) ii) If A is ot ivertible, the if x =1 { su Ax, y) y =1 } { } = su if Ax, y) y =1 x =1 = 0 314) 46
Proof i) a) dim H = 1 imlies that oe vector sas the sace ad Ax = ax for all x H ad some scalar a Let e H, e = 1 The e forms a comlete orthoormal set Thus { } { } { } if x =1 su Ax, y) y =1 = if x =1 su ax, y) y =1 = if l 1 =1 su al 1 e, l 2 e) l 2 =1 = a, ad { } su if Ax, y) y =1 x =1 { = su y =1 if ax, y) x =1 } { } = su if al 1e, l 2 e) = a l 2 =1 l 1 =1 b) Suose that A is ivertible This imlies that A is ijective ad that A 1 is bouded By theorem 311, A is bouded below Let if x =1 Ax = k > 0 The, for all x such that x = 1 su Ax, y) 217) = Ax k > 0 315) y =1 Therefore { } if x =1 su Ax, y) y =1 = if Ax = k > 0 x =1 Let us ow evaluate the right had side { } su if Ax, y) y =1 x =1 { = su y =1 if x, x =1 A y) } = 0 as dim H > 1 imlies that for each vector A y we ca fid a orthogoal vector x such that x = 1 ii) Suose ow that A is ot ivertible If A is ijective, Theorem 311 imlies that A is ot bouded below ie there exists a sequece {x } such that x = 1 47
for all ad lim Ax = 0 If A is ot ijective, there is e H, e = 1, such that Ae = 0 Set x = e for all The if x =1 { su Ax, y) y =1 } 217) = if Ax if Ax = 0 x =1 Hece, by Proositio 31, if A is ot ivertible, the miimax 314) holds 35 Alicatio Defiitio 313 [1] We say that a comlex fuctio Ω : H H C is a bouded biliear fuctioal o a Hilbert sace H if, for all x, y, z H, the followig coditios are satisfied: a) Ω α 1 x + α 2 y, z) = α 1 Ω x, z) + α 2 Ω y, z); b) Ω x, β 1 y + β 2 z) = β 1 Ω x, y) + β 2 Ω x, z); c) su x 1, y 1 Ω x, y) < The scalar roduct x, y) o H is a examle of a biliear fuctioal The orm of the biliear fuctioal Ω, is defied by Ω = Ω x, y) su Ω x, y) = su x =1, y =1 x,y H x y Thus Ω x, y) Ω x y for all x, y H Theorem 314 [1] Each biliear fuctioal Ω o a Hilbert sace H has a reresetatio of the form Ω x, y) = Ax, y) where A B H) ad A is uiquely defied by Ω Furthermore A = Ω 48
The miimax theorem ca be alied to a biliear fuctioal Ω o a Hilbert sace H as follows Cosider the miimax formula: if su Ω x, y) = su x =1 y =1 y =1 if x =1 Ω x, y) 316) Corollary 315 Let Ω be a bouded liear fuctioal o a Hilbert sace H ad let A be the corresodig oerator defied i Theorem 314 such that Ω x, y) = Ax, y) i) If A is ivertible ad a) dim H = 1, the the miimax formula 316) holds: if su Ω x, y) = su x =1 y =1 y =1 if x =1 where a is a scalar such that Ax = ax for all x H Ω x, y) = a, b) dim H > 1, the the miimax coditio 316) does ot hold: while if su Ω x, y) = if Ax = k > 0, x =1 y =1 x =1 su if y =1 x =1 Ω x, y) = 0 ii) If A is ot ivertible the the miimax coditio 316) holds: if su Ω x, y) = su x =1 y =1 y =1 if x =1 Ω x, y) = 0 36 Coclusio I this chater we studied miimax coditio for sequeces of bouded semiorms o a Hilbert sace H We foud that its validity deeds o comariso of the 49
bouded semiorms with the orm of the Hilbert sace H We illustrated the result with examle of bouded semiorms o the sace l 2 We also evaluated this miimax i reverse ad illustrated it with examles o l 2 We foud that, ulike the revious miimax theorem, the miimax i reverse holds i all cases Towards the ed of this chater we reseted a simle miimax formula for bouded oerators We foud that the miimax formula holds if the bouded oerator is ot ivertible ad it does ot hold if the oerator is ivertible ad dim H > 1 We comleted this chater with alicatio of the miimax coditio for oerators to bouded biliear fuctioals o H I the ext chater we study miimax theory for a secial class of comact oerators - the Schatte class oerators o a searable Hilbert sace H 50
Chater 4 Miimax ad Schatte ideals of comact oerators 41 Itroductio I this chater we cosider various miimax coditios for orms of comact oerators i Schatte ideals While i majority of cases the restrictios o oerators for which these coditios hold are straightforward, i oe case cosidered i Sectio 43 the fulfilmet of the miimax coditio deeds o a iterestig geometric roerty of a family of subsaces {L = P H} of a Hilbert sace aroximate itersectio of these subsaces Before we cosider these miimax coditios, let us recall mai cocets of theory of Schatte ideals that we will eed i this chater Let H be a searable Hilbert sace ad BH) be the C -algebra of all bouded oerators o H with oerator orm The set CH) of all comact oerators i BH) is the oly closed two-sided ideal of BH) [21, Corollary 11] However, BH) has may o-closed two-sided ideals By Calki theorem [21, Theorem 11], all these ideals of BH) lie i CH) Defiitio 41 [21, 68-70] A two-sided ideal J of BH) is called symmetrically ormed s ), if it is a Baach sace i some orm J ad AXB J A X J B for all A, B BH) ad X J 51
The most imortat class of s ideals - the class of Schatte ideals - is defied i the followig way [21, Theorem 71 ] For A CH), cosider the ositive oerator A = A A) 1/2 The oerator A is comact [43, Theorem 137], so that its sectrum σ A ) cotais 0 [30, Remark, 196], which is the oly limit oit of σ A ) Aart from 0, it cosists of coutably may ositive eigevalues of fiite multilicity see Theorem 235) Thus σ A )\{0} ca be writte as a o-icreasig sequece sa) = {s i A)} of eigevalues of A, takig accout of their multilicities Hece sa) belogs to the sace c 0 of all sequeces of real umbers covergig to 0 For each 0 <, cosider the followig fuctio o c 0 : ) 1/ φ ξ) = ξ i, where ξ = ξ 1,, ξ, ) c 0, i=1 ad the followig subset of comact oerators ) 1/ S = S H) = {A CH) : φ sa)) = s j A) < } 41) The all S are two-sided ideals of BH) [21] For each A S, cosider the orm ) 1/ A = φ sa)) = s j A) 42) For all 1 <, S are Baach -algebras with resect to the orms ad the adjoit oeratio as the ivolutio: if T S the T S [43, Theorem 136] Moreover, they are s ideals of BH) see [16, Lemma 6 c)] for the secod j j 52
statemet ad [21, Theorem 71] for the first statemet): AT B A T B ad T = T, 43) for all T S, A, B BH) These ideals are called Schatte ideals All Schatte ideals are searable algebras i the orm toology ad the ideal of all fiite rak oerators i BH) is dese i each of them [21, 92] Moreover [16, Lemma 9 a)], S q S, if q <, ad A A q if A S q 44) We also deote the ideal CH) of all comact oerators by S Note that [21, 27] A = A = su s j = s 1 45) Defiitio 42 [27, 28] [1, 61] [30, 164] A sequece {x } i a Hilbert sace H is said to coverge weakly to x H if lim x, y) = x, y) for all y H Let K be aother searable Hilbert sace Let {A } be a sequece of oerators i BH, K) It coverges to a bouded oerator A i the weak oerator toology wot), if A x, y) Ax, y) for all x H ad y K It coverges to A i the strog oerator toology sot), if Ax A x K 0 for all x H 53
If {x } coverges to x H i orm, the {x } weakly coverges to x If {A } uiformly coverges to a oerator A A A 0) the {A } sot A; if {A } sot A the {A } wot A We ca exted the orm to all oerators from BH), by settig A =, if A BH) ad A / S Thus A < if A S, ad A = if A / S, for [1, ) 46) All Schatte ideals S, [1, ), share the followig imortat roerty Theorem 43 [21, Theorem III51] Let [1, ) ad let a sequece {A } of oerators from S coverge to A BH) i the weak oerator toology If su A = M < the A S ad A M Theorem 43 imlies the followig result Corollary 44 [21, Theorem III52] Let a sequece {T } of oerators i B H) coverge to 1 H i the strog oerator toology Let [1, ) ad A B H) The followig coditios are equivalet i) A belogs to S ii) For some M 1 > 0, A satisfies su T AT = M 1 < 47) iii) For some M 2 > 0, A satisfies su T A = M 2 < 54
Proof As T x x 0, as, for all x H, it follows from the uiform boudedess ricile see, for examle, [16, Theorem II117] [30, Theorem 310]) that there is L > 0 such that su T < L i) iii) If A S the all T A S ad, by 43), T A T A L A Hece iii) holds for M 2 = L A iii) ii) As T A S, the oerators T AT also belog to S By 43), T AT T A T M 2 L = M 1 ii) i) Let 47) hold The sequece {T AT } coverges to A i sot Ideed, for each x H, Ax T AT x Ax T Ax + T Ax T AT x Ax T Ax + T A x T x 0, as, sice z T z 0 for all z H Hece {T AT } coverges to A i wot As T AT M 1 <, all oerators T AT belog to S Therefore it follows from Theorem 43 that A S ad A M 1 Corollary 44 is artially stated i Theorem III52 of [21, 87] but oly for mootoically icreasig sequece of fiite rak rojectios It should be oted that Corollary 44 does ot hold for =, that is, for S = CH) Ideed, let A be a bouded o-comact oerator The A / S 55
However, as the orm coicides with the usual oerator orm, we have that 47) holds, sice su T A = su T A su T A L A Theorem 45 [21, Theorem III63] Let {T } be a sequece of self-adjoit bouded oerators o H that coverges to 1 H i the strog oerator toology The, for each [1, ] ad for each A S, A T A 0 ad A T AT 0, as The above result meas that every sequece of self-adjoit bouded oerators o H that coverges to 1 H i the strog oerator toology is a aroximate idetity i all ideals S, [1, ] icludig S = CH)) Corollary 46 Let a sequece of self-adjoit bouded oerators {T } o H coverge to 1 H i the strog oerator toology Suose that su T 1 The, for each A B H) ad each [1, ], su T AT = A 48) ad lim T AT = A 49) Proof Let firstly A S It follows from 43) ad Theorem 45 that, T AT 43) T A T A ad lim T AT = A 56