On Some Properties of Tensor Product of Operators

Transcription

1 Global Joural of Pure ad Applied Matheatics. ISSN Volue 12, Nuber 6 (2016), pp Research Idia Publicatios O Soe Properties of Tesor Product of Operators Saleh Ora ad A. E. Sayed Departet of Matheatics, Faculty of Sciece, Taif Uiversity, Taif, Saudi Arabia. Abstract I this article we itroduced soe properties of operators o Hilbert spaces, specially the tesor product of two operators o bouded Hilbert space. Moreover we studied both iterior ad exterior tesor product of two operators o the Hilbert A -odule for give C*-algebra A. 1. INTRODUCTİON The approach i article is to study defiitio of the tesor product the Hilbert C*-odule over the tesor products of C*-algebras. We have used these defiitios to calculate the effects of liear uerical rage i Hilbert C*-odules. We cocluded the relatioship betwee the tesor product i this ad the classical tesor product i Hilbert spaces ad we proved that i the case of coplex ubers the tesor product of operators o Hilbert C*-odules coicide with the tesor product of operators o the Hilbert, we itroduced a brief suary of the tesor product of operators o bouded operator o Hilbert spaces, oe ca foud a rich aterial of this defiitio i [4,6]. Moreover the iterior ad the exterior tesor product of a operators o the bouded liear operators o Hilbert C*-odules was suarized ad we refer for this defiitios ad other results i [2,3,5,7]. 2. TENSOR PRODUCT OF BOUNDED OPERATORS ON HİLBERT SPACES Let H1 ad H2 be two vector spaces over the field K. deote by F( H1, H2 ) the fiite liear cobiatios such that

2 5140 Saleh Ora ad A. E. Sayed F (H 1, H 2 ) = { C i ( f i, g i ) C i ε k, f i ε H 1, g i ε H 2, i = 1, 2,..., } i=1 Is a vector space. Let N be the subspace of F (H 1, H 2 ) spaed by the eleets of the for a i b i f i g i ( a i f i, b i g i ) i=1 j=1 i=1 j=1 Is the quotiet space.the H 1 H 2 = F (H 1, H 2 )/ N. Is called the algebraic tesor product of the two vector spaces H 1 ad H 2. Ad the pair ( f i, g i ) is deoted by f i g j The liear cobiatio of eleets of the for a i b i f i g i ( a i f i ) ( b i g i ) = 0 i=1 j=1 So, I particular, we have i=1 j=1 a i b i f i g i = ( a i f i ) ( b i g i ) = 0 i=1 j=1 i=1 j=2 Defiitio 2.1 : Let ( H1, <, >1 ) ad ( H2,, 2 ) Be Hilbert spaces, the we ca defie the ier product i the for c i (f i g i ), C j (f j, g j ) i=1 j=1 It is a pre-hilbert space. The copletio of this pre-hilbert space is deoted by H 1 H 2 ad called the tesor product of the Hilbert spaces H 1 ad H 2. Deoted it by x y H1 H2 x H1, y H2 for typical eleets i H1 H 2. Defiitio 2.2: A operator T betwee two Hilbert spaces H 1 ad H 2 Is called bouded if there is K 0 Such that Tx K x for all x H i. That sallest such K is the or of T ad deoted T, such that. T = Sup{ Tx, x = 1, xεh 1 }. The set of all bouded operators o H is deoted by B(H). B(H) is copleted with the operator or ad satisfyig

3 O soe properties of tesor product of operators 5141 ST S. T for ay T, S (H ). Thus B ( H ) is Baach algebra, ad will deote by I the uit i B (H). Moreover if T ad S are two operators o B(H) there is a ivolutio * satisfyig the followig. i) T, y = x, T y. ii)( S + T ) = S + T, ( T ) = α T ad (T ) = T ii) T = T ad T T = T 2, for T B(H), α C I fact B(H) is defied C*- algebra. If H 1 ad H 2 are Hilbert spaces, there are atural extesio of operators i B (Hi), I = 1, 2, to the operators o H 1 H 2 if T B (H 1 ), defie T I B (H 1 H 2 ) Siilarly, for S B ( (H 2 ), I S Ad T T I ad S B(H 1 ) ad B(H 2 ) so we get i geeral ( T I )(x y ) = T x y is defied by ( I S )(x y ) = x Sy I S are isoetries. The iages of is deoted by B(H 1 ) I ad I B(H 2 ) respectively ad If S1, T1 B (H 1 ) ad S 2, T 2 B(H 2 ) We ca defied S1 S2 B (H 1 H 2 ) by (S1 S2)( x y = S 1 x S 2 y The where is defid by S 1 S 2 = (S 1 I )( I S 2 ) = (I S 2 )(S 1 I), (S 1 + T 1 ) S 2 = (S 1 S 2 ) + (T 1 S 2 ), S 1 (S 2 + T 2 ) = (S 1 S 2 ) + (S 1 T 2 ), (S 2 S 2 )(T 1 T 2 ) = S 1 T 1 S 2 T 2, (S 1 S 2 ) = S 1 S 2 ad S 1 S 2 = S 1. S 2. T x, y = x, Ty. by

4 5142 Saleh Ora ad A. E. Sayed Theore 3.1 : If T 1, T 2,, T are liearly idepedet i B(H 1 ) ad S 1, S 2,, S are liearly idepedet i B(H 2 ). The T 1 S 1 + T 2 S T S = 0 if ad oly if T 1 = T 2 = T = 0. Proof : SiceT 1, T 2,, T are liearly idepedet. This gives for soe vectors η i, ζ i H i, The for arbitrary vectors x ad y i H 2, The we have r S 1 x, y = T j ζ i, η i S j x, y j=1 i =1 r = T j S j, (ζ i x ), (ζ i xy ) j=1 i =1 = 0 So S 1 x, y = 0 ad this give S 1 = 0 Siilarly we ca get S 2 = S 2 = = S = 0. Theore 4.2 : let T i B (H i ), i = Ad T 1 T 2 T Is a operator i H 1 H 2 T the T 1 T 2 T is oral opertater if T i are oral opertators. Proof : It is suffices to prove that for T 1 T 2. let T 1 T 2 is oral operator i B ( H 1 H 2 ) the (T 1 T 2 ). (T 1 T 2 ) = (T 1 T 2 )((T 1 T 2 ). ie T 1 T 1 T 2 T 2 T 1 T 1 T 2 T 2 = 0 if T 1 T 1 ad T 1 T 1 are liearly idepedet. the T 2 T 2 = T 2 T 2 = 0, so T 2 = 0 Coutraductio, So T 1 T 1 = = α T 1 T 1 ad siilaraly T 2 T 2 = β T 2 T 2 ad T 1 2 = Sup{ T 1 x, T 1 x, x = 1}. = Sup{ rt 1 x, T 1 x, x = 1} = Sup{r T 1 x, T 1 x, x = 1} = r T 1 2 Sice T 1 = T 1 ther fore r = 1. Ad siilarly α = 1. So T 1 T 1 = T 1 T 1 ad T 2 T 2 = T 2 T 2 The T 1 ad T 2 are oral operators, the other had is easy to prove.

5 O soe properties of tesor product of operators C*-ALGEBRAS AND HİLBERT C*-MODULES Defiitio 3.1: A*-algebra is a algebra A together with a ivolutio. A Baach *-algebra is a Baach algebra A together with a isoetric ivolutio. (That is, we isist that a = a for all a A ) A algebra hooorphis φ: A B betwee *-algebras is called a *-hooorphis if φ(a )= φ(a) for all a A. Defiitio 3.2 A Baach algebraa is a Baach space together with a or copatible algebra structure, aely: forallx, y A, xy x y. If A has a ultiplicativeuit (deotedby e or 1), we say that the algebra A is uital. I this case, e = e 2 e 2, ad so e = 0 iplies that e 1. By scalig the or, we assue that e = 1. Exaple 3.3 The set (C(X),. ) of cotiuous fuctios o a copact Hausdorff space X itroduced i Exaple becoes a Baach algebra uder poitwise ultiplicatio of fuctios. That is, for f, g C(X),. ), we set (fg)(x) = f(x)g(x) forall x X. Exaple 3.4 :LetX be a Baach Space, the the Baach space B(X) with the operatör or A = Sup { Ax x = 1} is defied a Baach Algebra. Defiitio 3.5 A Baach*-algebra is called a C algebra if a a = a 2 a A. Exaple 3.6.If X is a locally copact Hausdorff space, thea = C 0 (X) is a C -algebra whe equipped with theivolutio give by coplex cojugatio. for all As we shall see, every coutative C -algebra arises as i Exaple 2.9. For the caoical o coutative exaple, we eed the followig. Exaple 3.7: Coplex ubers are with the or γ = γ is C -algebra ore details about C*-algebras i [1,4,6]. Exaple 3.8 :B(H) = {T: H H T is bouded} with the operator or T = Sup{ Tx x H, x = 1} defied as a C -algebra Reark 3.9 I geeral, if a C*- algebra A does ot have a idetity eleet, it is possible to apped oe as follows: Cosider the algebra A = A C1 with ultiplicatio give by (a, λ) (b, ν) = (ab + aν + bλ, λν). We defie a or o A via (a, λ1) = a + λ. The wehave (a, λ) (b, ν) = ab + aν + bλ + λν a b + a ν + b λ + λ ν = ( a + λ ) ( b + ν )

6 5144 Saleh Ora ad A. E. Sayed = ( (a, λ) ) ( (b, ν) ) It is clear that the ebeddig of A ito A is liear ad isoetric, ad that A sits iside of A as a closed ideal. Defiitio 3.10 : let A be a C a lgebra. A A pre Hilbert - odule is a coplex vector space E which a right A-odule with the ap <, > E E A. Which is liear i the secod variable ad satisfy the followig relatios, for all a A ad x, y E, α, β C i) x, x 0 ad if x, x = 0 iplies x = 0 ii) x, y = y, x iii) x, ya = y, x a iv) x, α y + β = α x, y + β x, z It is easy to check that the ultiplicatio ad the right A - odule structure are copatible i the sese that λ(xa ) = (λx )a = x (λ a ) for x E, a A, λ C E is called right pre- A odule. Defiitio 3.11: The or of a eleet x E is defied as x = x, x. The or o E satisfy i) x, y x. y, ii) x + y x + y, iii) x, y 2 x, x. y, y ad vi) xa x. a for x, y E ad a A If the pre-hilbert odule is coplete with the respect or above it is said to be Hilbert odule over A. I fact i the case right actio the pre- Hilbert odule is called Right Hilbert odule. Exaple 3.12 : C algebra A itself is a Hilbert A- odule with the ier product a, b = a b Let E1, ad E2 are Hilbert A-odules deoted by L A (E 1, E 2 ) The space of all adjoitable operators T E 1 E 2 Satisfy Tx, y = x, T y for all x E 1 ad y E 2. These operators are liear bouded operators fro E 1 to E 2 ad have a or T = sup{ Tx, x = 1}.

7 O soe properties of tesor product of operators 5145 Theore 3.13 : L A (E) the set of all bouded opertator o the Hilbert A odule E is C algebra. Proof : it is straight forward to check that L A (E) is a algebra ad ST S. T holds, ad oe oe ca show that T = T, Ad TX, Tx = x, T Tx T Tx T T, x = 1 This yields T 2 T T ad it is clear that T T T 2 ad so So L A (E) is C algebra. Defiitio : let L A (E) be the set of all boud opretor o E. Let T L A (E) L A (E) Deote by W A (T) = { Tx, x A : x = 1}, the uerrical rage of T i E. Lea 3.14 : i) W A (T ) = W A (T) ii) WA (T) = 1 proof : i) W A (T ) = { T x, x x = 1 = { x, Tx, x = 1} = { Tx, x x = 1} = W(T). ii) WA (I) = { Ix, x x = 1} = { x, x = 1 = x 2 = 1}. Defiitio 3.15: Iterior tesor product. Let E be a Hilbert A-odule, F a hilbert B- odule Let φ A L B (F)be a * - hoorphis. The φ akes F is a left A odule, ad the actio give by a, y φ(a)y ( a A, y F ) Ad we ca for the algebraic tesor product of E ad F over A, E A F which is a right B- odule. <, > E A F E A F B satisfies x 1 y 1, x 2 y 2 = y 1, φ ( x 1, x 2 )y 2. We ca set N = {z E A F z, z = 0} The N is B subodule ad we cosider the quistio E A F /N ad the quotiet ab q: E A F E A F /N.

8 5146 Saleh Ora ad A. E. Sayed The E A F / N is a right B odule. Ad the ier product q(x), q(y) = x, y. Defiitio 3.15: Exterior i tesor product. Let E be a Hilbert A odule ad F a Hilbert B-odule. the algebraic tesor product E C F is a right odule over the algebraic tesor product A C B such that (x y)(a b) = xa yb, x E, y F, a A, b B. We ca defied A B valued ier product E F by the ap, E F x E F A B satisfy. x 1 y 1, x 2 y 2 = x 1, x 2 y 1, y 2. Thus E F is a pre-a B odule. For ore detale see [2,3]. Defiitio 3.16: let T L A (E)ad S L A (F), The T S L A ( E F) where E F is the extorior tesor product of E ad F, the we defie the uerical rage of the tesor product of T S L A ( E F) as W A (T S) = { (T S) (x y ), x y ) : x y = 1 }. Theore 3.17: Let T ad S be two operators o L A ( E ), the W A (T + S) W A (T) + W A (S) Proof : It is straightfowrward sice the uerical rage is liear. ACKNOWLEDGMENTS The authors are grateful to Taif Uiversity Saudi Arabia for it fiacial support for research uber 3426/435/1. REFRENCES [1] B. Blackadar, Operator algebras. Theory of C* -algebras ad vo Neua algebras, Ecyclopaedia of Matheatical Scieces 122. Operator Algebras ad No-Coutative Geoetry 3. Berli: Spriger (2006). [2] I. Kaplasky, Modules over operator algebras, Aerica Joural of Matheatics 75(4)(1953), [3] E. C. Lace, Hilbert C -odules, Lodo Math, Soc. Lect. Note Series, Vol 210, Cabridge Uiversity Press, (1995).

9 O soe properties of tesor product of operators 5147 [4] G. J. Murphy. C*- algebras ad Operatio Theory. Acadeic Press, [5] W. L. Paschke, Ier product odules over C*-algebras, Trasactios of the Aerica Matheatical Society 182(1973), [6] G. K. Pederse, 1979, C* - algebras ad autoorphis groups. Acadeic Press, Lodo. [7] K. Those, Hilbert C* -odules, K-theory ad C* -extesios, Aarhus Uiversitet, 38. Aarhus: Aarhus Uiversitet, Mate

10 5148 Saleh Ora ad A. E. Sayed