Hölder Type Inequalities Involving Tracy-Singh Products and Khatri-Rao Products of Hilbert Space Operators
Keywords:Hölder inequality, Tracy-Singh product, Khatri-Rao product, Hilbert space operator
This paper generalizes the famous Hölder type inequalities for positive real numbers to positive operators on an arbitrary complex Hilbert space. We use appropriate integral representations of certain operator-monotone functions to deduce the concavity and convexity of certain maps involving Tracy-Singh products of operators. These results lead to Hölder type inequalities for operators concerning Tracy-Singh products, Khatri-Rao products, Tracy-Singh sums, and Khatri-Rao sums. In particular, we obtain Cauchy-Schwarz type inequalities for operators involving Tracy-Singh products and Khatri-Rao products.
Z. Al-Zhour, New Hölder-type Inequalities for the Tracy-Singh and Khatri-Rao products of positive matrices, Intelligent Control and Automation 3(3), 50 – 54 (2012), DOI: 10.5897/IJCER12.013.
T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (1979), 203 – 241, DOI: 10.1016/0024-3795(79)90179-4.
J. I. Fujii, Operator-valued inner product and operator inequalities, Banach J. Math. Anal. 2(2) (2008), 59 – 67, DOI: 10.15352/bjma/1240336292.
F. Hiai and D. Petz, Introduction to Matrix Analysis and Applications, Springer, New Delhi (2014).
J. L. W. V. Jensen, Sur les fonctions convexes et les inégalités entre les valeurs moyennes, Acta Math. 30 (1906), 175 – 193, DOI: 10.1007/BF02418571.
A. Ploymukda and P. Chansangaim, Algebraic, order, and analytic properties of Tracy-Singh sums for Hilbert space operators, Songklanakarin J. Sci. Technol. 41(4) (2019), 727 – 733.
A. Ploymukda, P. Chansangiam and W. Lewkeeratiyutkul, Algebraic and order properties of Tracy-Singh products for operator matrices, J. Comput. Anal. Appl. 24(4) (2018), 656 – 664.
A. Ploymukda, P. Chansangiam and W. Lewkeeratiyutkul, Analytic properties of Tracy-Singh products for operator matrices, J. Comput. Anal. Appl. 24(4) (2018), 665 – 674.
A. Ploymukda and P. Chansangaim, Concavity and convexity of several maps involving Tracy-Singh products, Khatri-Rao products, and operator-monotone functions of positive operators, ScienceAsia, 45 (2019), 194 – 201, DOI: 10.2306/scienceasia1513-1874.2019.45.194.
A. Ploymukda and P. Chansangaim, Khatri-Rao products and selection operators, J. Comput. Anal. Appl. 27(2) (2019), 316 – 325.
A. Ploymukda and P. Chansangiam, Khatri-Rao products of operator matrices acting on the direct sum of Hilbert spaces, J. Math. 7 pages (2016), DOI: 10.1155/2016/8301709.
A. Ploymukda and P. Chansangaim, Khatri-Rao sums for Hilbert space operators, Songklanakarin J. Sci. Technol. 40(3) (2018), 595 – 601, DOI: 10.14456/sjst-psu.2018.76.
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