HILBERT-SCHMIDT AND TRACE CLASS OPERATORS MICHAEL WALTER Let H 0 be a Hilbert space. We deote by BpHq ad KpHq the algebra of bouded respective compact operators o H ad by B fi phq the subspace of operator of fiite rak. We write if two spaces are isometrically isomorphic. The space of bouded sequeces with idex set J is deoted by l 8 pjq, its (closed) subspace of zero sequeces by c 0 pjq ad the subspace of sequeces with fiite support by c fi pjq. The space of (square) summable sequeces is writte as l 2 pjq ad l 1 pjq, respectively. 1. Itroductio Recall that we have the followig hierarchy classic sequece spaces: c fi pnq Ď l 1 pnq Ď l 2 pnq Ď c 0 pnq Ď l 8 pnq They are Baach spaces (for c fi pnq) ad commutative algebras; l 2 pnq eve is a Hilbert space. Similarly, we have the followig chai of operator algebras: B fi phq Ď? Ď? Ď KpHq Ď BpHq They are Baach spaces (except for B fi phq) ad algebras, although o-commutative i geeral. The followig propositio shows that we ca i a sese iterprete these operator algebras as the o-commutative aaloga of the respective sequece spaces. 1 Propositio. For ay orthoormal system pe q i H we have a isometric algebra homomorphism Φ : l 8 Ñ BpHq, pa q ÞÑ x ÞÑ a xx, e ye with Φ 1 pb fi phqq c fi ad Φ 1 pkphqq c 0. Proof. We oly prove that Φ is well-defied ad a isometry: a xx, e ye 2 a 2 xx, e y 2 ď pa q 2 l xx, e 2 y 2 ď pa q 2 l 2 x 2 ad ř a xe m, e ye a m, hece Φppa qq pa q l 2. It is thus atural to ask the followig questios: (1) What operator algebras correspod to l 1 pnq ad l 2 pnq? (2) Which familiar results from the theory of sequece spaces geeralize to the o-commutative case? 2. Hilbert-Schmidt operators We defie the space of Hilbert-Schmidt operators as B 2 phq : ta P BpHq : A 2 ă 8u d A 2 : Ae i 2 where pe i q ipi is a ONB of H. This is a ormed space. A easy calculatio shows that this defiitio does ot deped o the choice of basis: 2 Lemma. Let A P BpHq ad let pe i q ipi, pf j q jpj be two ONBs. The: Ae i 2 A f j 2 P r0, 8s ipi jpj ipi Date: Last compiled Jauary 28, 2011; last modified February 8, 2010 or later. BASED ON [Nee96] 1
2 MICHAEL WALTER Proof. Suppose the first limit exist. The by Fubii we have Ae i 2 xae i, f j y 2 ipi ipi jpj xa f j, e i y 2 A f j 2, jpj ipi jpj hece the secod limit exists ad agrees. The followig facts follow easily from the precedig. 3 Propositio. Let A P B 2 phq. The: (i) A 2 A 2 (ii) A ď A 2 (iii) B 2 phq is a operator ideal i BpHq, i.e. BpHqB 2 phqbphq Ď B 2 phq Proof. (i) Lemma 2. (ii) Let ɛ ą 0. Take e P H such that e 1 ad Ae ą A ɛ, exted to a ONB pe i q. The A 2 2 i Ae i 2 ě Ae 2 ą p A ɛq 2 (iii) It is clear that B 2 phq is a left ideal; (i) shows that it is a right ideal. 4 Theorem. pb 2 phq, 2 q is a Hilbert space with ier product xa, By 2 : i xb Ae i, e i y i xae i, Be i y ad B fi phq is a dese subspace. Proof. Cosider the mappig From the calculatios Ψ : c fi pi ˆ Iq Ñ B 2 phq, δ pq ÞÑ x, e i ye j Ψppa qq 2 ď Ψppa qq 2 2 a x, e i ye j 2 2 a 2 pa q 2 l 2 we see that Ψ has a cotiuous extesio l 2 pi ˆ Iq Ñ BpHq which is a surjective isometry oto B 2 phq. Thus the latter is also a Hilbert space with dese subspace Ψpc fi pi ˆ Iqq B fi phq. The formula for the ier product is easily obtaied usig the polarizatio idetity. 5 Corollary. B 2 phq Ď KpHq Proof. Theorem 4 ad Propositio 3, (ii). Ay Hilbert-Schmidt operator A P B 2 phq Ď KpHq ca be writte as a series A a x, e yf with pa q P c 0 pnq ad orthoormal systems pe q, pf q. We ca easily calculate its orm from ay such represetatio: 6 Propositio. c A 2 a 2 pa q l 2 Thus a compact operator is a Hilbert-Schmidt operator if ad oly if its coefficiets are i l 2 pnq. Fially we will reveal the itimate coectio betwee the Hilbert-Schmidt operators o H ad the tesor product of H with its dual.
HILBERT-SCHMIDT AND TRACE CLASS OPERATORS 1 3 7 Propositio. The space of Hilbert-Schmidt operator is aturally isometrically isomorphic to the tesor product H b H via Φ : H b H Ñ B 2 phq, λ b f ÞÑ λf Proof. The mappig Φ is iduced by the biliear map pλ, fq ÞÑ λf, hece well-defied. Choose a ONB pe i q of H. The x, e i y b e j is a ONB of the tesor product H b H, ad Φp a x, e i y b e j q 2 2 a 2 a x, e i y b e j 2 a x, e i ye j 2 2 shows that Φ is a isometry. Thus Φ is also surjective sice its rage icludes the dese set of fiite-rak operators. Now apply the bouded iverse theorem. 3. Trace class operators We defie the space of trace class (or uclear) operators to be This is a ormed space. Let us first collect some facts about this space. 8 Propositio. Let A P B 1 phq. The: (i) A 1 A 1 (ii) A 2 ď A 1 B 1 phq : ta P B 2 phq : A 1 ă 8u A 1 : supt xa, By 2 : B P B 2 phq, B ď 1u (iii) B 1 phq is a operator ideal i BpHq, i.e. BpHqB 1 phqbphq Ď B 1 phq (iv) B 2 phqb 2 phq Ď B 1 phq Proof. (i) We have xa, By 2 xb, A y 2 sice both sides defie ier products iducig the same orm (apply the polarizatio idetity). This i tur implies the claim. (ii) This follows from A ď A 2. (iii) I view of (i) we oly have to show that B 1 phq is a left ideal; this follows readily from xca, By 2 xa, C By 2. (iv) Let A, B, C P B 2 phq ad B ď 1. The hece CA 1 ď C 2 A 2. xca, By 2 xa, C By 2 ď A 2 C B 2 A 2 B C 2 ď A 2 B C 2 ď A 2 C 2, We defie the trace of a trace class operator A P B 1 phq to be trpaq : ipixae i, e i y where pe i q ipi is a ONB of H. Note that this coicides with the usual defiitio of the trace if H is fiitedimesioal. The followig lemma shows that the defiitio make sese. 9 Lemma. The series coverges absolutely ad it is idepedet from the choice of basis. i xae i, e i y i Proof. Choose λ i P C such that xae i, e i y λ i xae i, e i y ad λ i 1 (i P I). The for every fiite subset I 0 Ď I we have the followig estimate: λ i xae i, e i y i λ i xa, x, e i ye i y 2 xa, i This implies absolute covergece. λ i x, e i ye i y 2 ď A 1 i λ i x, e i ye i ď A 1
4 MICHAEL WALTER If pf j q jpj is ay other ONB we have xae i, e i y xae i, f i yxf i, e i y xf i, e i yxe i, A f i y i j,i xf i, A f i y xaf i, f i y, j j hece the trace is idepedet from the particular choice of basis. We ow collect some facts about the trace which resemble the fiite-dimesioal case. 10 Propositio. (i) tr P B 1 phq 1 with tr 1 (ii) trpabq trpbaq for A P B 1 phq, B P BpHq ad A, B P B 2 phq, respectively Proof. (i) By the proof of the precedig lemma we have tr ď 1. Equality follows by cosiderig a orthogoal projectio. (ii) If A P B 1 phq is Hermitia ad B P BpHq we ca take a ONB of eigevectors pe i q with Ae i : λ i e i for real eigevalues λ i P R. The trpabq i xabe i, e i y i xbe i, Ae i y i xbae i, e i y trpbaq If A P B 1 phq is a geeral trace class operator we ca still write it as a sum A B ` ic with Hermitia B, C P B 1 phq. The claim the follows from the complex biliearity of pa, Bq ÞÑ trpabq ad pa, Bq ÞÑ trpbaq. For A, B P B 2 phq the claim follows from trpabq i xa, B y i xabe i, e i y xb, A y xbae i, e i y trpbaq Let us write a trace class operator A P B 1 phq as a series A a x, e yf with pa q P c 0 pnq ad orthoormal systems pe q, pf q. Agai it is easy to calculate its orm ad trace from this represetatio: 11 Propositio. A 1 trpaq a pa q l 1 a xf e y Thus a compact operator is a trace class operator if ad oly if its coefficiets are i l 1 pnq. Proof. We oly show the first equality; the secod oe is immediate from the defiitio of tr. pďq For ay B P B 2 phq with B ď 1 we have xa, By 2 ď a xx, e yf, By 2 ď a xf, Be y ď a, hece A 1 ď ř a. pěq Choose b P C such that a a b ad b 1 ( P N) ad defie N B N : b x, e yf 1
HILBERT-SCHMIDT AND TRACE CLASS OPERATORS 2 5 Clearly B N P B 2 phq ad B N ď 1. Hece A 1 ě xa, B N y 2 xae, B N e y N N a b a Ò 1 1 a It follows that we ca approximate ay trace class operator usig fiite rak operators: 12 Corollary. B fi is a dese subspace of pb 1 phq, 1 q. Proof. We have N 8 A a x, e yf 11 1 a Ñ 0 as N Ñ 8. 1 N We ca also deduce that every trace class operator is the product of two Hilbert-Schmidt operators: 13 Propositio. B 2 phqb 2 phq B 1 phq Proof. pďq was proved i Propositio 8, (iv). pěq Defie? a x, e yf B C? a x, e ye The B ad C are Hilbert-Schmidt operators, ad A BC. Note that is a cotiuous pairig sice we have B 1 phq ˆ BpHq Ñ C, pa, Bq ÞÑ trpabq trpabq ď AB 1 ď A 1 B This pairig iduces the followig two isometric isomorphisms. 14 Theorem. B 1 phq KpHq 1 ad B 1 phq 1 BpHq Proof. (1) We show that is a isometric isomorphism. B 1 phq Ñ KpHq 1, B ÞÑ trp Bq Liearity is obvious. It is almost by defiitio of 1 that the mappig is a isometry. Hece it remais to show surjectivity. Let ϕ P KpHq 1. The for all A P B 2 phq we have ϕpaq ď ϕ A ď ϕ A 2, hece ϕ B2pHq P B 2 phq 1. Take the uique B P B 2 phq such that ϕ B2pHq x, By 2 xb, y 2 trp B q From this we see that the cotiuity of ϕ implies that B is of trace class, ad desity of B 2 phq Ď KpHq shows that B is a preimage of ϕ. (2) We show that is a isometric isomorphism. BpHq Ñ B 1 phq 1, B ÞÑ trp Bq Agai, it is obvious that the mappig is a liear isometry. We show surjectivity. Let ϕ P B 1 phq 1. Sice for e, f P H x, eyf 1 e f we see that the mappig f ÞÑ ϕpx, eyfq is i H 1. Hece there is a uique ϕ e P H such that xf, ϕ e y ϕpx, eyfq @f P H
6 MICHAEL WALTER Table 1. Compariso of sequece ad operator spaces Sequece spaces Operator spaces c fi pnq dese i l 1 pnq, B fi phq dese i B 1 phq, l 2 pnq ad c 0 pnq B 2 phq ad KpHq l 1 pnq l 2 pnql 2 pnq pa q ÞÑ ř a P l 1 pnq 1 B 1 phq B 2 phqb 2 phq tr P B 1 phq 1 c 0 pnq 1 l 1 pnq KpHq 1 B 1 phq l 1 pnq 1 l 8 pnq B 1 phq 1 BpHq ad ϕ e ď ϕ e. Thus B : H Ñ H, e ÞÑ ϕ e defies a bouded operator. Ad the calculatio ϕpx, eyfq xf, Bey xx, eyf, By 2 trpx, eyfb q together with desity of B fi phq Ď B 1 phq shows that B is a preimage of ϕ. 15 Corollary. pb 1 phq, 1 q is a Baach space. 4. Summary Propositios 6 ad 11 show that the algebras of Hilbert-Schmidt ad trace class operators are the atural ocommutative aaloga of l 1 pnq ad l 2 pnq, respectively. That is, we have the followig chais which correspod i the sese of Propositio 1: c fi pnq Ď l 1 pnq Ď l 2 pnq Ď c 0 pnq Ď l 8 pnq B fi phq Ď B 1 phq Ď B 2 phq Ď KpHq Ď BpHq I table 1 we have summarized some familiar facts about sequece spaces together with their o-commutative couterparts (which we have proved i the precedig). Refereces [Nee96] Karl-Herma Neeb. Skript zur Vorlesug Spektral- ud Darstellugstheorie. http://www.mathematik.tu-darmstadt.de/ fbereiche/alggeofa/staff/eeb/skripte/spektraltheorie-ss96.pdf, 1996.