Research Article Khatri-Rao Products for Operator Matrices Acting on the Direct Sum of Hilbert Spaces

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1 Joural of Mathematics Volume 2016, Article ID , 7 pages Research Article Khatri-Rao Products for Operator Matrices Actig o the Direct Sum of Hilbert Spaces Aro Ploymukda ad Pattrawut Chasagiam Departmet of Mathematics, Faculty of Sciece, Kig Mogkut s Istitute of Techology Ladkrabag, Chalogkrug Rd, Bagkok 10520, Thailad Correspodece should be addressed to Pattrawut Chasagiam; pattrawutch@kmitlacth Received 2 July 2016; Revised 20 September 2016; Accepted 18 October 2016 Academic Editor: Ralf Meyer Copyright 2016 A Ploymukda ad P Chasagiam This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited We itroduce the otio of Khatri-Rao product for operator matrices actig o the direct sum of Hilbert spaces This otio geeralizes the tesor product ad Hadamard product of operators ad the Khatri-Rao product of matrices We ivestigate algebraic properties, positivity, ad mootoicity of the Khatri-Rao product Moreover, there is a uital positive liear map takig Tracy- Sigh products to Khatri-Rao products via a isometry 1 Itroductio I matrix theory, there are various matrix products which are of iterest i both theory ad applicatios, such as the Kroecker product, Hadamard product, ad Khatri-Rao product; see, for example, [1 3 Deote by M m, (C) the set of m-by- complex matrices ad abbreviate M, (C) to M (C) Recall that the Kroecker product of A[a ij M m, (C) ad B M p,q (C) is give by A B [a ij B ij M mp,q (C) (1) The Hadamard product of A, B M m, (C) is defied by the etrywise product A B [a ij b ij M m, (C) (2) Now, let A ad B be complex matrices partitioed ito blocks A ij ad B ij for each i, j (the sizes of A ij ad B ij may be differet) The, the Khatri-Rao product [4 of A ad B is defied by A B [A ij B ij ij (3) Whe A ad B are opartitioed (ie, each has oly oe block), their Khatri-Rao product is just their Kroecker product If A ad B are etrywise partitioed (ie, each block is a 1 1matrix), the their Khatri-Rao product is their Hadamard product Iterestig algebraic, order, ad aalytic properties of this product were studied i the literature; see, for example, [5 12 Their applicatios i statistics, computer sciece, ad related fields ca be see, for example, i [13, 14 The tesor product of Hilbert space operators is a atural extesio of the Kroecker product to the ifiitedimesioal settig Let H, H, K, adk be Hilbert spaces Recall that the tesor product of two operators A:H H ad B:K K is the uique bouded liear operator from H K ito H K such that, for all x H ad y K, (A B) (x y)ax By (4) I this paper, we geeralize the tesor product of operators to the Khatri-Rao product of operator matrices actig o a direct sum of Hilbert spaces We ivestigate fudametal properties of this operator product Algebraically, this product is compatible with the additio, the scalar multiplicatio, the adjoit operatio, ad the direct sum of operators By itroducig suitable operator matrices, we ca prove that there is a uital positive liear map takig the Tracy-Sigh product A Bto the Khatri-Rao product A BHece,the Khatri-Rao product ca be viewed as a geeralizatio of the Hadamard product of operators Moreover, positivity, strict

2 2 Joural of Mathematics positivity, ad operator orderigs are preserved uder the Khatri-Rao product Our result exteds well-kow results for Khatri-Rao products of complex matrices (see [4, 9, 15, 16) This paper is orgaized as follows I Sectio 2, we provide some prelimiaries about Tracy-Sigh products for operators These facts will be used i Sectios 4 ad 5 I Sectio3,weitroducetheKhatri-Raoproductforoperator matrices ad deduce its algebraic properties Sectio 4 explais how the Khatri-Rao product ca be regarded as a geeralizatio of the Hadamard product Sectio 5 discusses positivity ad mootoicity of Khatri-Rao products 2 Prelimiaries o Tracy-Sigh Products for Operators Throughout, let H, H, K, adk be complex separable Hilbert spaces Whe X ad Y are Hilbert spaces, deote by B(X, Y) the Baach space of bouded liear operators from X ito Y, adabbreviateb(x, X) to B(X) Ifaoperator A B(H) satisfies Ax, x > 0, wewritea>0forselfadjoit operators A, B B(H),wewriteA Bto mea that A Bis a positive operator, while A>Bmeas that A B > 0 Decompose H H K K j1 m q l1 p k1 H j, H i, K l, K k, where all H j, H i, K l,adk k are Hilbert spaces For each j, l, letm j : H j H ad N l : K l K be the caoical embeddigs For each i ad k, letp i : H H i ad Q k : K K k be the orthogoal projectios Two operators A B(H, H ) ad B B(K, K ) ca thus be represeted uiquely as operator matrices A[A ij m, i,j1, (5) B[B kl p,q k,l1, (6) where A ij P i AM j B(H j, H i ) ad B kl Q k BN l B(K l, K k ) for each i, j, k, adl We defie the Tracy-Sigh product of A ad B to be the bouded liear operator from,q j,l1 H j K l to m,p i,k1 H i K k expressed i a block-matrix form A B [[A ij B kl kl ij (7) Basic properties of the Tracy-Sigh product are listed below Lemma 1 The Tracy-Sigh product (A, B) A B is a biliear map for operators It is positive i the sese that if A 0ad B 0,theA B 0 3 Compatibility of Khatri-Rao Products with Algebraic Operatios I this sectio, we defie the Khatri-Rao product for operator matrices ad show that this product is compatible with certai algebraic operatios of operators From ow o, fix the followig orthogoal decompositios of Hilbert spaces: H H K K j1 m j1 m H j, H i, K j, K i That is, we fix how to partitio ay operator matrix i B(H, H ) ad B(K, K ) We ow exted the Khatri-Rao product of matrices [4 to that of operators o a Hilbert space Defiitio 2 Let A [A ij m, i,j1 B(H, H ) ad B [B ij m, i,j1 B(K, K ) be operators partitioed ito matrices accordig to decompositio (8) We defie the Khatri-Rao product of A ad B to be the bouded liear operator from j1 H j K j to m H i K i represeted by the block-matrix (8) A B[A ij B ij m, (9) i,j1 If both A ad B are 1 1block operator matrices, the A Bis A BWheH i K i C ad H j K j C for all i, j, the Khatri-Rao product is the Hadamard product of complex matrices Next, we shall show that the Khatri-Rao product of two liear maps iduced by matrices is just the liear map iduced by the Khatri-Rao product of these matrices Recall that, for each A M m, (C) ad B M p,q (C), theiduced maps, L A : C C m, L B : C q C p, x Ax, y By, (10) are bouded liear operators We idetify C C q with C q together with the caoical biliear map (x, y) x y for each (x, y) C C q Lemma 3 For ay A M m, (C) ad B M p,q (C),oehas L A L B L A B (11)

3 Joural of Mathematics 3 Proof Recall that the Kroecker product of matrices has the followig property (see, eg, [3): Similarly, we obtai property (17) Sice (αa) ij αa ij for all i, j,weget (A B) (C D) AC BD (12) provided that all matrix products are well defied It follows that, for ay x C ad y C q, (L A L B )(x y)l A (x) L B (y) (αa) B[(αA ij ) B ij ij [α(a ij B ij ) ij α(a B) Similarly, A (αb) α(a B) (21) L A (x) L B (y) Ax By (A B) (x y) (A B) (x y) L A B (x y) (13) The uiqueess of tesor products implies that L A L B L A B Propositio 4 For ay complex matrices A[A ij ad B [B ij partitioed i block-matrix form, oe has L A L B L A B (14) Proof Recall that the (i, j)th block of the matrix represetatio of L A is L Aij By Lemma 3, we obtai L A L B [L Aij L Bij ij [L Aij B ij ij L A B The ext result states that the Khatri-Rao product is biliear ad compatible with the adjoit operatio Propositio 5 Let A B(H, H ) ad B, C B(K, K ) be operator matrices, ad let α CThe, (A B) A B, (15) A (B+C) A B+A C, (16) (B+C) AB A+C A, (17) (αa) Bα(A B) A (αb) (18) Proof Sice A [A ji ij ad B [B ji ij,weobtai By property (15), the self-adjoitess of operators is closed uder takig Khatri-Rao products; that is, if A ad B are self-adjoit, the so is A BTheextpropositio shows that, i order to compute the Khatri-Rao product of operator matrices, we ca freely merge the partitio of each operator Propositio 6 Let A [A ij m, i,j1 B(H, H ) ad B [B ij m, i,j1 B(K, K ) be operator matrices represeted accordig to decompositio (8) We merge the partitio of A to be A[A kl r,s k,l1,wherer, s are give atural umbers such that r mad s Here, each operator A kl is of m k l block i which the (g, h)th block of A kl is the (u, V)th block of A,where r k1 s l1 g, { k 1 u k 1 { m i +g, k>1, { m k m, h, { l 1 V l 1 { j +h, l>1, { j1 l (22) (A B) [(A ij B ij ) ij [A ji B ji ij A B (19) Similarly, we repartitio B [B kl r,s k,l1, where each operator B kl is of m k l block i which the (g, h)th block of B kl is the (u, V)th block of BThe, The fact that (B + C) ij B ij +C ij for all i, j together with the left distributivity of the tesor product over the additio implies A (B+C) [A ij (B ij +C ij ) ij [(A ij B ij )+(A ij C ij ) ij A B+A C (20) A B[A kl B kl kl A 11 B 11 A 1s B 1s [ d [ A r1 B r1 A rs B rs That is, each (k, l)th block of A Bis just A kl B kl (23)

4 4 Joural of Mathematics Proof Write A B [C kl r,s k,l1,whereckl is m k l block operator matrix such that the (g, h)th block of C kl is the (u, V)th block of A B We kow that the (u, V)th block of A Bis A uv B uv The, A 11 B 11 A 11 B 11 C 11 [ d [ A m1 1 B m1 1 A m1 1 B m1 1 A 11 A 11 B 11 B 11 [ d [ d [ A m1 1 A m1 1 [ B m1 1 B m1 1 A 11 B 11 (24) Similarly, we have C kl A kl B kl for all k 1,,r ad l1,,s Recall that the direct sum of A i B(H i, H i ),,,, is defied to be the operator matrix A A 2 0 A 1 A [ d (25) [ 0 0 A The ext result shows that the Khatri-Rao product is compatible with the direct sum of operators Propositio 7 For each i 1,,,letA i B(H i, H i ) ad B i B(K i, K i ) be compatible operator matrices The, ( A i ) ( B i ) Proof It follows directly from Propositio 6 (A i B i ) (26) I summary, the Khatri-Rao product is compatible with fudametal algebraic operatios for operators 4 The Khatri-Rao Product as a Geeralizatio of the Hadamard Product I this sectio, we explai how the Khatri-Rao product ca be viewed as a geeralizatio of the Hadamard product To do this, we costruct two isometries which idetify which blocks of the Tracy-Sigh product we eed to get the Khatri- Rao product Fix a coutable orthoormal basis E for H Recall that the Hadamard product of A ad B i B(H) is defied to be the operator A Bi B(H) such that (A B) e, e Ae, e Be, e (27) for all e E More explicitly, it was show i [15 that A BU (A B) U, (28) where U:H H H is the isometry defied by Ue e e for all e E WheH C ad E is the stadard ordered basis of C, the Hadamard product of two matrices reduces to the etrywise product (2) We ow exted selectio matrices i [9 to selectio operators Fix a ordered 4-tuple (H, H, K, K ) of Hilbert spaces edowed with decompositio (8) For each r 1,,m, cosider the operator matrix E r [E (r) gh m,m g,h1 : m H m i K i H r K i, (29) where E (r) gh is the idetity operator if ghradthe others are zero operators Similarly, for s 1,,, we defie the operator matrix F s [F (s) gh, g,h1 : j1 H j K j j1 H s K j, (30) where F (s) gh is the idetity operator if ghsadthe others are zero operators Now, costruct E 1 Z 1 [, [ E m F 1 Z 2 [ [ F (31) We call Z 1 ad Z 2 selectio operators associated with the ordered tuple (H, H, K, K )NoticethatZ 1 depeds oly o the ordered tuple (H, K ) ad how we decomposed H ad K TheoperatorZ 2 depeds o (H, K) ad how we decomposed H ad K For istace, a ordered tuple (H, H, K, K ) with decompositios H H 1 H 2 H 3, H H 1 H 2, K K 1 K 2 K 3, K K 1 K 2 has the followig selectio operators: Z 1 [ I H 0 1 K 1, 0 I H 2 K 2 Z 2 [ I H1 K [ 0 I H2 K 2 0 [ 0 0 I H3 K 3 (32) (33)

5 Joural of Mathematics 5 IthecaseofH H ad K K,costructio(31)gives Z 1 Z 2 Z (34) If (Z 1,Z 2 ) is the ordered pair of selectio operators associated with the ordered tuple (H, H, K, K ) with decompositios give by (8), the (Z 2,Z 1 ) is the ordered pair of selectio operators associated with the ordered collectio (H, H, K, K) with the same decompositios Lemma 8 Let Z 1 ad Z 2 be selectio operators defied by (31) The, for,2, (i) Z i Z i I;thatis,Z i is a isometry; (ii) 0 Z i Z i I Proof A direct computatio shows that Z 1 Z 1 I ad Z 2 Z 2 IWekowthatE i E i is a m mblock operator matrix which cosists oly of zero ad idetity operators More precisely, the (i, i)th block of E i E i is the idetity operator ad E i E j 0for i jthe, E 1 E 1 E 1 E 2 E 1 E m [ E 2 E Z 1 Z 1 [[[[[ 1 E 2 E 2 E 2 E m d [ E m E 1 E m E 2 E m E m E 1 E E 2 E 2 0 [ d [ 0 0 E m E m (35) Sice E i E i I for all i 1,,m,wehaveZ 1 Z 1 I Similarly, Z 2 Z 2 I Next, we relate the Khatri-Rao ad the Tracy-Sigh product of operators Theorem 9 For ay operator matrices A B(H, H ) ad B B(K, K ),oehas A BZ 1 (A B) Z 2, (36) where Z 1 ad Z 2 are the selectio operators defied by (31) If H H ad K K, A B(H) ad B B(K),the A BZ (A B) Z, (37) where Z is the selectio operator defied by (34) Proof Let B(i) deote the ith colum of B for i 1,, The, we have Z 1 (A B) Z 2 A 11 B A 1 B F 1 [ [E 1 E [[[ m d [ [ A m1 B A m B [ F (A 11 B)F 1 + +(A 1 B)F [ [E 1 E [[[ m [(A m1 B)F 1 + +(A m B)F A 11 B(1) A 1 B() [ [E 1 E [[[ m [ A m1 B(1) A m B() A 11 B 11 A 1 B 1 [ d A B [ A m1 B m1 A m B m (38) If H H ad K K,theZ 1 Z 2 ad (36) becomes (37) We metio that Theorem 9 is a extesio of both [9, Theorem 3ad result(28)i[15 Remark 10 If we partitio A ad B ito row operator matrices, we have A B(A B) Z 2 (39) If both A ad B are colum operator matrices, the A BZ 1 (A B) (40) Comparig (28) ad (37), Theorem 9 shows that the Khatri-Rao product ca be regarded as a geeralizatio of the Hadamard product Recall that a map Φ betwee two C -algebras is said to be positive if Φ preserves positive elemets The map Φ is uital if Φ preserves the multiplicative idetity Corollary 11 There is a uital positive liear map, Φ:B ( H i K j ) B ( H i K i ) (41) i,j1 such that Φ(A B) A B for ay A B(H) ad B B(K) Proof Defie Φ(X) Z XZ, wherez is the selectio operator defied by (37) i Theorem 9 The map Φ is clearly liear ad positive The map Φ is uital sice Z is a isometry (Lemma 8) Corollary 11 provides a atural way to derive operator iequalities cocerig Khatri-Rao products from existig iequalities for Tracy-Sigh products The ext result exteds [16, Corollary 3 to the case of Khatri-Rao ad Tracy-Sigh products of operators Corollary 12 Let AA 1 A ad BB 1 B be operators i B(H) ad B(K), respectively The, Z (A B) (A B) Z, (A B) ZZ(A B) (42)

6 6 Joural of Mathematics Proof Usig the fact that E i E i XX i XE i E i ad E i E j X 0XE i E j if i j,wherexx 1 X,wecompute ZZ (A B) Proof Applyig Propositio 5 ad Theorem 13 yields (A 1 B 1 ) (A 2 B 2 ) A 1 B 1 A 2 B 1 +A 2 B 1 A 2 B 2 (45) E 1 E 1 0 A 1 B 0 [ d [ d [ 0 E E [ 0 A B E 1 E 1 (A 1 B) 0 [ d [ 0 E E (A B) (A 1 B)E 1 E 1 0 [ d [ 0 (A B)E E A 1 B 0 E 1 E 1 0 [ d [ d [ 0 A B [ 0 E E (A B) ZZ By applyig Theorem 9, we get Z (A B) Z ZZ (A B) Z (A B) ZZ (A B) Z Similarly, (A B)Z Z(A B) 5 Positivity ad Mootoicity of Khatri-Rao Products (43) (44) (A 1 A 2 ) B 1 +A 2 (B 1 B 2 ) 0 Thus, A 1 B 1 A 2 B 2 Now, we will develop the result of [9, Theorem 6 tothe case of Khatri-Rao product of operators Theorem 15 Let A B(H) ad B B(K) be operator matrices If A>0ad B>0,the A B>0 Proof The strict positivity of A ad the spectral theorem imply the existece of a icreasig sequece (H ) 1 of closed subspaces of H such that, for each N, Ax, x 1 x 2 (46) for each x H LetP be the orthogoal projectio oto H for each N There are similar subspaces K ad orthogoal projectios Q for the operator B The, for each N,wehaveA (1/)P ad B (1/)Q ad hece A B 1 2 P Q (47) by Corollary 14 Sice the uio of the subspaces H i H ad of the subspaces K i K is dese, it follows that, for ay z H K,thereism N for which (P m Q m )z, z > 0Hece, (A B) z, z 1 m 2 (P m Q m )z,z >0 (48) This shows that A B>0 Corollary 16 Let A 1,A 2 B(H) ad B 1,B 2 B(K) If A 1 >A 2 >0ad B 1 >B 2 >0,theA 1 B 1 >A 2 B 2 I this sectio, we show that the Khatri-Rao product preserves positivity ad strict positivity It follows that operator orderigs are preserved uder Khatri-Rao products Theorem 13 Let A B(H) ad B B(K) be operator matrices If A 0ad B 0,theA B 0 Proof It follows from the positivity of the Tracy-Sigh product (Lemma 1) ad Theorem 9 The ext result provides the mootoicity of Khatri-Rao product which is a extesio of [9, Theorem 5 to the case of operators Corollary 14 Let A 1,A 2 B(H) ad B 1,B 2 B(K) If A 1 A 2 0ad B 1 B 2 0,theA 1 B 1 A 2 B 2 Proof The proof is similar to that of Corollary 14 Istead of Theorem 13, we apply Theorem 15 Fially, we metio that, by usig the results i this paper, we ca develop further operator idetities/iequalities parallel to matrix results for Khatri-Rao products Competig Iterests The authors declare that they have o competig iterests Ackowledgmets This work was supported by the Thailad Research Fud The secod author would like to thak the Thailad Research Fud for the fiacial support

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