scholarly journals From Matrix to Operator Inequalities

2012 ◽  
Vol 55 (2) ◽  
pp. 339-350 ◽  
Author(s):  
Terry A. Loring
Keyword(s):  
Technical Condition ◽  
Monotone Functions ◽  
Bounded Operators ◽  
Operator Inequalities ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Operator Monotone ◽  
Noncommutative Polynomials ◽  
Definition Of

AbstractWe generalize Löwner's method for proving that matrix monotone functions are operator monotone. The relation x ≤ y on bounded operators is our model for a definition of C*-relations being residually finite dimensional.Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices.Applications are shown regarding norms of exponentials, the norms of commutators, and “positive” noncommutative ∗-polynomials.

scholarly journals Reverse and improved inequalities for operator monotone functions

2020 ◽  
Vol 74 (2) ◽  
pp. 19
Author(s):  
Sever Dragomir
Keyword(s):  
Power Function ◽  
Hilbert Spaces ◽  
Monotone Functions ◽  
Operator Inequalities ◽  
Operator Monotone ◽  
Operator Monotone Functions

In this paper we provide several refinements and reverse operator inequalities for operator monotone functions in Hilbert spaces. We also obtain refinements and a reverse of Lowner-Heinz celebrated inequality that holds in the case of power function.

scholarly journals Positivity, Betweenness, and Strictness of Operator Means

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Pattrawut Chansangiam
Keyword(s):  
Fixed Point ◽  
Binary Operation ◽  
Point Property ◽  
Monotone Functions ◽  
Operator Equations ◽  
Operator Inequalities ◽  
Borel Measures ◽  
Operator Monotone ◽  
Operator Means ◽  
Operator Monotone Functions

An operator mean is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, the transformer inequality, and the fixed-point property. It is well known that there are one-to-one correspondences between operator means, operator monotone functions, and Borel measures. In this paper, we provide various characterizations for the concepts of positivity, betweenness, and strictness of operator means in terms of operator inequalities, operator monotone functions, Borel measures, and certain operator equations.

Some operator inequalities involving operator monotone functions

2021 ◽  
Vol 166 ◽  
pp. 102938
Author(s):  
Hosna Jafarmanesh ◽  
Maryam Khosravi ◽  
Alemeh Sheikhhosseini
Keyword(s):  
Monotone Functions ◽  
Operator Inequalities ◽  
Operator Monotone ◽  
Operator Monotone Functions

scholarly journals A Survey on Operator Monotonicity, Operator Convexity, and Operator Means

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Pattrawut Chansangiam
Keyword(s):  
Measure Theory ◽  
Integral Representations ◽  
Differential Analysis ◽  
Monotone Functions ◽  
Operator Inequalities ◽  
Borel Measures ◽  
Operator Monotone ◽  
Operator Means ◽  
Operator Monotone Functions ◽  
The Relationship

This paper is an expository devoted to an important class of real-valued functions introduced by Löwner, namely, operator monotone functions. This concept is closely related to operator convex/concave functions. Various characterizations for such functions are given from the viewpoint of differential analysis in terms of matrix of divided differences. From the viewpoint of operator inequalities, various characterizations and the relationship between operator monotonicity and operator convexity are given by Hansen and Pedersen. In the viewpoint of measure theory, operator monotone functions on the nonnegative reals admit meaningful integral representations with respect to Borel measures on the unit interval. Furthermore, Kubo-Ando theory asserts the correspondence between operator monotone functions and operator means.

Spectral synthesis and residually finite-dimensional algebras

2017 ◽  
Vol 16 (10) ◽  
pp. 1750200 ◽  
Author(s):  
László Székelyhidi ◽  
Bettina Wilkens
Keyword(s):  
Spectral Analysis ◽  
Abelian Group ◽  
Theoretical Approach ◽  
Abelian Groups ◽  
Spectral Synthesis ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Analysis And Synthesis ◽  
Spectral Analysis And Synthesis

In 2004, a counterexample was given for a 1965 result of R. J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Since then the investigation of discrete spectral analysis and synthesis has gained traction. Characterizations of the Abelian groups that possess spectral analysis and spectral synthesis, respectively, were published in 2005. A characterization of the varieties on discrete Abelian groups enjoying spectral synthesis is still missing. We present a ring theoretical approach to the issue. In particular, we provide a generalization of the Principal Ideal Theorem on discrete Abelian groups.

scholarly journals Kwong matrices and operator monotone functions on $(0,1)$

2014 ◽  
Vol 5 (1) ◽  
pp. 121-127 ◽  
Author(s):  
Juri Morishita ◽  
Takashi Sano ◽  
Shintaro Tachibana
Keyword(s):  
Monotone Functions ◽  
Operator Monotone ◽  
Operator Monotone Functions
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