An operator equality involving a continuous field of operators and its norm inequalities

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1 Available olie at Liear Algebra ad its Alicatios 49 (008) A oerator equality ivolvig a cotiuous field of oerators ad its orm iequalities Mohammad Sal Moslehia a, b,, Fuzhe Zhag c, d a Deartmet of Pure Mathematics, Ferdowsi Uiversity of Mashhad, P.O. Box 59, Mashhad 9775, Ira b Cetre of Excellece i Aalysis o Algebraic Structures (CEAAS), Ferdowsi Uiversity of Mashhad, Ira c Farquhar College of Arts ad Scieces, Nova Southeaster Uiversity, 330 College Aveue, Fort Lauderdale, FL 3334, USA d College of Mathematics ad Systems Sciece, Sheyag Normal Uiversity, Sheyag, Liaoig Provice 0034, Chia Received 5 Jauary 008; acceted 9 Jue 008 Available olie July 008 Submitted by C.-K. Li Abstract Let A be a C -algebra, be a locally comact Hausdorff sace equied with a robability measure P ad let (A t ) t be a cotiuous field of oerators i A such that the fuctio t A t is orm cotiuous o ad the fuctio t A t is itegrable. he the followig equality icludig Boucher itegrals holds A t A s dp dp A t dp A t dp. (0.) his equality is related both to the otio of variace i statistics ad to a characterizatio of ier roduct saces. With this oerator equality, we reset some uiform orm ad Schatte -orm iequalities. 008 Elsevier Ic. All rights reserved. AMS classificatio: Primary: 47A6; Secodary: 46C5, 47A30, 5A4 Keywords: Bouded liear oerator; Characterizatio of ier roduct sace; Hilbert sace; -Norm; Norm iequality; Schatte -orm; Cotiuous filed of oerators; Boucher itegral Corresodig author. Address: Deartmet of Mathematics, Ferdowsi Uiversity of Mashhad, P.O. Box 59, Mashhad 9775, Ira. addresses: moslehia@ferdowsi.um.ac.ir, moslehia@ams.org (M.S. Moslehia), zhag@ova.edu (F. Zhag) /$ - see frot matter ( 008 Elsevier Ic. All rights reserved. doi:0.06/j.laa

2 60 M.S. Moslehia, F. Zhag / Liear Algebra ad its Alicatios 49 (008) Itroductio ad relimiaries May iterestig characterizatios of ier roduct saces have bee itroduced (see, e.g. []). It was show by Rassias [7] that a ormed sace X (with orm ) is a ier roduct sace if ad oly if for ay fiite set of vectors x,...,x X, x i x j x i x j. (.) his equality is of fudametal imortace i the study of ormed saces ad ier roduct saces sice it reveals a basic relatio betwee the two sorts of saces. From statistical oit of view, let (X, μ) be a robability measure sace ad f be a radom variable, i.e., f is a elemet of L (X, μ). With the variace of f defied by Var(f ) E( f E(f) ), where E(f) X f dμ deotes the exectatio of f,(.) resembles (or vice versa) the wellkow equality Var(f ) E( f ) E(f). Let B(H) be the algebra of all bouded liear oerators o a searable comlex Hilbert sace H edowed with ier roduct, ad the oerator orm. We deote the absolute value of A B(H) by A (A A) /.Forx,y H, the rak oe oerator x y is defied o H by (x y)(z) z, y x. Let A be a C -algebra ad let be a locally comact Hausdorff sace. A field (A t ) t of oerators i A is called a cotiuous field of oerators if the fuctio t A t is orm cotiuous o.ifμ(t) is a Rado measure o ad the fuctio t A t is itegrable, oe ca form the Bocher itegral A tdμ(t), which is the uique elemet i A such that ( ) ϕ A t dμ(t) ϕ(a t ) dμ(t) for every liear fuctioal ϕ i the orm dual A of A; cf. [5, Sectio 4.]. Let A B(H) be a comact oerator ad let 0 <<. he Schatte -orm (-quasiorm) for < (0 <<) is defied by A (tr A ) /, where tr is the usual trace fuctioal. Clearly, for, q > 0, A q A q/ /. For >0, the Schatte -class, deoted by C, is defied to be the two-sided ideal i B(H) of those comact oerators A for which A is fiite. I articular, C ad C are the trace class ad the Hilbert Schmidt class, resectively. For more iformatio o the theory of the Schatte -classes, the reader is referred to [,8]. Sice C is a Hilbert sace uder the ier roduct A, B tr(b A), it follows from (.) that if A,...,A C, the A i A i. (.)

3 M.S. Moslehia, F. Zhag / Liear Algebra ad its Alicatios 49 (008) I this aer, we establish a geeral oerator versio of equality (.), from which we deduce a extesio of (.). We reset some iequalities cocerig various orms such as Schatte - orms that form atural geeralizatios (i iequalities) of the idetity (.) (see, e.g. [6]). It seems that the iequalities related to Schatte -orms are useful i oerator theory ad mathematical hysics ad are iterestig i their ow right.. Mai results We begi by establishig a oerator versio of equality (.) ivolvig cotiuous fields of oerators ad itegral meas of oerators. heorem.. Let A be a C -algebra, be a locally comact Hausdorff sace equied with a robability measure P ad let (A t ) t be a cotiuous field of oerators i A such that the fuctio t A t is orm cotiuous o ad the fuctio t A t is itegrable. he A t A s dp(s) dp(t) A t dp(t) A t dp(t) (.) Proof A t A s dp(s) dp(t) ( ) ( ) A t A s dp(s) A t A r dp(r) dp(t) ( ) A t dp(t) A t A r dp(r) dp(t) (( ) ) A s dp(s) A t dp(t) ( ) + A s dp(s) A r dp(r) dp(t) ( ) A t dp(t) A s dp(s) A r dp(r) ( by dp(t) ad A t dp(t) ( A t dp(t) ) ) A t dp(t) A t dp(t). Corollary.. Let A,...,A B(H). he (i) for ay set of oegative umbers t,...,t with t i, t i A i t j t i A i t j ; (.)

4 6 M.S. Moslehia, F. Zhag / Liear Algebra ad its Alicatios 49 (008) (ii) A i A i. (.3) Proof. (i) ake {,,} ad P({i}) t i i heorem.. (ii) Put t i / i (i). As a immediate cosequece of (.), we get the followig kow AM-M oerator iequality; cf. [9, heorem 7]. Corollary.. Let A,...,A B(H). he for ay set of oegative umbers t,...,t with t i, t i A i t i A i. he followig result comares the mea of the squares of the oerators to the square of the mea, ad gives some bouds of their differece. Corollary.3. Let A,...,A B(H) be ositive oerators such that 0 m i I A i M i I for some oegative scalars m i ad M i ad all i, ad let t,...,t be oegative umbers such that t i. he where ad t i βi I α i max { β i mi { ( ) t i A i t i A i M i M i t i m i, m i t i m i, m i t i αi I, } t i M i } t i M i. Proof. It is sufficiet to otice that for each A i, by the fuctioal calculus, mi M i t i m i, m i t i M i A i t j max M i t i m i, m i t i M i ad use (.).

5 M.S. Moslehia, F. Zhag / Liear Algebra ad its Alicatios 49 (008) By a -orm o a subsace D of B(H) we mea a orm for which there exists a orm defied o a subsace D such that A A A (A D) (we imlicitly assume that A A D if ad oly if A D). he oerator orm ad (for < ) are examles of -orms o B(H); see, e.g., [,. 89] or [3]. Corollary.4. Let be a -orm o a subsace D of B(H), ad let t,...,t be oegative umbers such that t i. he for ay A,...,A D with A i 0 for i/ j, ti A i t i A i t j + t j. Proof ti A i ti A i t i A i t i A i t j + t j t i A i t j + t j. Remark.. he oerators A i actig o a Hilbert sace havig the orthogoal roerty A i 0 for i/ j are ot ucommo. For istace, let (e i ) be a orthogoal family (ot cotaiig zero) i H ad defie the oerators A i : H H by A i e i e i e i,i. he A i s are ositive oerators i B(H) with A i e i for all i ad A i e i,e j for all i, j (for details see [4]). I the settig of Hilbert saces, the kow equality (.) ca be roved directly. I what follows we show that a extesio of it ca also be obtaied from equality (.). Corollary.5. Let x,...,x H. he t i x i t j x j t i x i t j x j. Proof. Let e be a o-zero vector of H ad set A i x i e. It follows from the elemetary roerties of rak oe oerators ad equality (.) that

6 64 M.S. Moslehia, F. Zhag / Liear Algebra ad its Alicatios 49 (008) t i x i t j x j e e t i x i t j x j e t i A i t j t i A i t j t i x i e t j x j e t i x i t j x j e e from which we coclude the result. From ow o we restrict ourselves to (.3). A secial case (for X C) of equality (.)is z i z j z i z j (z,...,z C). his equality is i tur a secial case (whe A diag(z,...,z ) ad tr([a ij ]) : a ii )of the ext equality cocerig the usual ormalized trace fuctioal. heorem.. Let tr(a) tr(a)/tr(i) be the ormalized trace o M (C). he A tr(a) A tr(i) tr(a). Proof A tr(a) tr A tr(a) tr(i) I (( ) ( tr A tr(a) tr(i) I A tr(a) ) ) tr(i) I tr(a A) tr(a )tr(a) tr(i) tr A tr(i) tr(a) tr(i) A tr(i) tr(a). tr(a)tr(a) tr(i) + tr(a) tr(i) tr(i) (by tr(a ) tr(a))

7 M.S. Moslehia, F. Zhag / Liear Algebra ad its Alicatios 49 (008) We eed the followig lemma which ca be deduced from [3, Lemma 4,9,. 0] (see also [6, Lemma.]). Lemma.. Let A,...,A be ositive oerators i C for some >0. (i) If 0 <<, the A i A i (ii) If <, the A i A i A i. A i. Note that the commutative versio of Lemma. for scalars follows from the well-kow Hölder iequality (see, e.g. [,. 88]). he ext theorem is our secod mai result. It ca be regarded as a geeralizatio of (.) i iequalities. heorem.3. Let A,...,A C for some >0. (i) If 0 <<, the A i (ii) If <<, the A i A i A i.. Proof. Let 0 <<. he A i + A i / + / A i / + / (by the secod iequality of Lemma.(i)) / /

8 66 M.S. Moslehia, F. Zhag / Liear Algebra ad its Alicatios 49 (008) / A i / A i / / (by equality (.3) (by the first iequality of Lemma.(i)) A i. his roves the first art of the theorem. By a similar argumet, oe ca rove the secod art. he ext theorem may be comared to heorem.3. It ca also be viewed as a variace of (.). heorem.4. Let A,...,A C for some >0. he A i A i for 0 <<; ad A i for <. A i Proof. Let 0 <<. he A i + A i + / A i + A i / (by equality (.3) ( / / A i /) / (by the secod iequality of Lemma.(i))

9 M.S. Moslehia, F. Zhag / Liear Algebra ad its Alicatios 49 (008) ( A i ) / A i. ( Ai ) / (by a scalar versio of Lemma.(ii)) his roves the first art of the theorem. he secod art of the theorem follows from a similar argumet. Ackowledgemet he authors would like to exress their gratitude to the referee for his/her very useful suggestios. he secod author thaks the NSU Farquhar College of Arts ad Scieces for a Mii-grat. Refereces [] D. Amir, Characterizatios of Ier Product Saces i Oerator heory: Advaces ad Alicatios, vol. 0, Birkhüser Verlag, Basel, 986. [] R. Bhatia, Matrix Aalysis, Sriger-Verlag, New York, 997. [3] R. Bhatia, F. Kittaeh, Norm iequalities for artitioed oerators ad a alicatio, Math. A. 87 (4) (990) [4] S.S. Dragomir, Some Schwarz tye iequalities for sequeces of oerators i Hilbert saces, Bull. Austral. Math. Soc. 73 () (006) 7 6. [5] F. Hase, G.K. Pederse, Jese s oerator iequality, Bull. Lodo Math. Soc. 35 (003) [6] O. Hirzallah, F. Kittaeh, M.S. Moslehia, Schatte -orm iequalities related to a characterizatio of ier roduct saces. Available from: <arxiv:080.76v>. [7] h.m. Rassias, New characterizatios of ier roduct saces, Bull. Sci. Math. 08 () (984) [8] B. Simo, race Ideals ad their Alicatios, Cambridge Uiversity Press, Cambridge, 979. [9] F. Zhag, O the Bohr iequality of oerators, J. Math. Aal. Al. 333 (007) 64 7.