Recent Developments of Mond-Pečarić Method in Operator Inequalities

Mićić Hot, Jadranka and Fujii, Masatoshi and Pečarić, Josip and Seo, Yuki (2012) Recent Developments of Mond-Pečarić Method in Operator Inequalities. = Recent Developments of Mond-Pečarić Method in Operator Inequalities. Monographs in inequalities . Element, Zagreb, -. ISBN 978-953-197-574-2

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Abstract

In Chapter 1 we give a very brief and quick review of the basic facts about a Hilbert space and (bounded linear) operators on a Hilbert space, which will recur throughout the book. In Chapter 2 we tell the history of the Kantorovich inequality, and describe how the Kantorovich inequality develops in the field of operator inequalities. In such context, the method for convex functions established by Mond and Pe\v{;c};ari\'{;c}; (commonly known as "the Mond- Pečarić method") has outlined a more complete picture of that inequality in the field of operator inequalities. We discuss ratio and difference type converses of operator versions of Jensen's inequality. These constants in terms of spectra of given self- adjoint operators have many interesting properties and are connected with a closed relation, and play an essential role in the remainder of this book. In Chapter 3 we explain fundamental operator inequalities related to the Furuta inequality. The base point is the Lowner-Heinz inequality. It induces weighted geometric means, which serves as an excellent technical tool. The chaotic order log A >= log B is conceptually important in the late discussion. In Chapter 4 we study the order preserving operator inequality in another direction which differs from the Furuta inequality. We investigate the Kantorovich type inequalities related to the operator ordering and the chaotic one. In Chapter 5 as applications of the Mond-Pečarić method for convex functions, we discuss inequalities involving the operator norm. Among others, we show a converse of the Araki-Cordes inequality, the norm inequality of seve\-ral geometric means and a complement of the Ando-Hiai inequality. Also, we discuss Holder's inequality and its converses in connection with the operator geometric mean. In Chapter 6 we define the geometric mean of n- operators due to Ando-Li-Mathias and Lowson-Lim. We present an alternative proof of the power convergence of the symmetrization procedure on the weighted geometric mean due to Lawson and Lim. We show a converse of the weighted arithmetic-geometric mean inequality of n-operators. In Chapter 7 we give some differential-geometrical structure of operators. The space of positive invertible operators of a unital C*-algebra has the natural structure of a reductive homogenous manifold with a Finsler metric. Then a pair of points A and B can be joined by a unique geodesic A #_t B for t in [0, 1] and we consider estimates of the upper bounds for the difference between the geodesic and extended interpolation paths in terms of the spectra of positive operators. In Chapter 8 we give some properties of Mercer's type inequalities. A variant of Jensen's operator inequality for convex functions, which is a generalization of Mercer's result, is proved. We show a monotonicity property for Mercer's power means for operators, and a comparison theorem for quasi-arithmetic means for operators. In Chapter 9 a general formulation of Jensen's operator inequality for some non-unital fields of positive linear mappings is given. Next, we consider different types of converse inequalities. We discuss the ordering among power functions in a general setting. We get the order among power means and some comparison theorems for quasi- arithmetic means. We also give a refined calculation of bounds in converses of Jensen's operator inequality. In Chapter 10 we give Jensen's operator inequality without operator convexity. We observe this inequality for n-tuples of self-adjoint operators, unital n-tuples of positive linear mappings and real valued convex functions with conditions on the operators bounds. In the present context, we also give an extension and a refinement of Jensen's operator inequality. As an application we get the order among quasi-arithmetic operator means. In Chapter 11 we observe some operator versions of Bohr's inequality. Using a general result involving matrix ordering, we derive several inequalities of Bohr's type. Furthermore, we present an approach to Bohr's inequality based on a generalization of the parallelogram theorem with absolute values of operators. Finally, applying Jensen's operator inequality we get a generalization of Bohr's inequality.

Item Type: Book
Keywords (Croatian): Jensen's inequality, Kantorovich inequality, Mond - Pečarić method, operator inequality, convex function
Subjects: NATURAL SCIENCES > Mathematics
Divisions: 1500 Chair of Mathematics
Date Deposited: 22 Sep 2016 11:32
Last Modified: 23 May 2017 11:27
URI: http://repozitorij.fsb.hr/id/eprint/6907

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