SOME GENERALIZATIONS OF CHEBYSHEV TYPE INEQUALITIES INVOLVING THE HADAMARD PRODUCT IN L^p SPACES CONSIST OF OPERATORS VALUE FUNCTIONS

Subject Areas : Statistics

Rudin Teimourian ¹ , Amir ghasem Ghazanfari ²

1 - Department of Mathematics, Lorestan University, P. O. Box 465, Khoramabad,Iran.
2 - Department of MathematicsLorestan UniversityP. O. Box 465, Khoramabad, Iran.

Received: 2019-09-11 Accepted : 2020-01-09 Published : 2021-02-19

Keywords: نامساوی عملگری, نامساوی چبیشف, ضرب هادامارد, نامساوی شوارتس,

Abstract :

Let B(H) denotes the C*-algebra of all bounded linear operators on a complex Hilbert space H together with the operator norm. Suppose A is a Banach *- subalgebra of B(H) , Ω a compact Hausdorff space equipped with a Radon measure μ and α:Ω→[0,1] is an integrable function. We first introduce the space L^p consists of all operator-valued functions from Ω to A which have finite norm related to a L^p-norm. Next, it is proved that if p and q are conjugate exponents, for every two elements belongs to L^p and L^q with almost synchronous property for the Hadamard product, then we will have a new operator Chebyshev type inequality involving the Hadamard product.Also using some properties of positive linear functional "tr", we introduce a semi-inner product for square integrable functions of operators in L^2. Using the obtained results, we prove the Schwarz and Chebyshev type inequalities dealing with the Hadamard product.

References:

[1]   T. Furuta, J. Mićić Hot, J. J. Pečarić and Y. Seo, Mond-J. Pečarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.

[2]   G.P.H. Styan, Hadamard product and multivariate statistical analysis, Linear Algebra Appl. 6 (1973), 217-240.

[3]   P.L. Chebyshev,O približennyh vyraženijah odnih integralov čerez drugie, in: Soobŝćenija i protokoly zasedanĭ Matemmati-českogo občestva pri Imperatorskom Har’kovskom Universitete, No. 2, 1882, pp. 93-98; Polnoe sobranie sočinenĭ P.L. Chebyshev, MoskvaLeningrad, 1948, pp. 128-131.

[4]   G. Grüss, Über das Maximum des absoluten Betrages von , Math. Z. 39 (1934), 215-226.

[5]   S. S. Dragomir, Operators inequalities of the Jensen, Čebyŝev and Grüss type, SpringerBriefs in Mathematics, Springer, New York, 2012. MR2866026.

[6]   M.S. Moslehian and R. Rajić, Grüss inequality for n-positive linear maps, Linear Algebra Appl. 433 (2010), 1555-1560.

[7]   S. Barza, L.-E. Persson and J. Soria, Sharp weighted multidimensional integral inequalities of Chebyshev type, J. Math. Anal. Appl. 236(2) (1999) 243-253.

[8]   M. W. Alomari, Pompeiu –Čebyŝev type inequalities for selfadjoint operators in Hilbert spaces, Adv. Oper. Theory 3 (2018), no 3, 459-472. MR3795049.

[9]   R. Drnovšck, A. Peperko, Inequalitieson the spectral radius and the operator norm of Hadamard products of positive operators on sequence, Banach J. Math. Anal. 10(2016), no. 4, 800-814. Mr3548627.

[10]    M.S. Moslehian and M.  Bakharad, Chebyshev type inequality for Hilbert space operators, J. Math. Anal. Appl. 420 (2014), 737-749.

[11]    J. Diestel and JR. J. J. Uhl, Vector measures, with a foreword by B. J. Pettis, Mathematical Surveys, No. 15, Amer. Math. Soc., Providence, R.I., 1977.

[12]    J. Mikusiński, The Bochner Integral, Birkhauser Verlag Basel, 1978.

[13]    F. Bahrami, A. Bayati Eshkaftaki and S. M. Manjegani, Operator-valued Bochner integrable functions and Jensen's inequality, Georgian Math. J.20, (2013), 625-640.

[14]    X. Li and W. Wu, Operator Jensen's inequality on C*-algebras, Acta Mathematica Sinica, English Series Jan. 30(1) (2014), 35-50.

[15]    J. G. Murphy, C*-algebras and operator theory, Academic Press, Boston, 1990.