Inequalities for the extended positive part of a von Neumann algebra related to operator-monotone and operator-convex functions

Dinh, Trung Hoa; Tikhonov, O. E.; Veselova, L. V.

doi:10.1215/20088752-2018-0040

Cited by 5 publications

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“…A real function f defined on an interval I is said to be operator convex for bounded operators if for all self-adjoint bounded operators A, B with spectrum in I, for each λ ∈ [0, 1], we have for bounded operators are also operator convex for unbounded operators [28].…”

Section: Definitionmentioning

confidence: 99%

“…The set of operator convex functions can be even smaller for unbounded operators than it is for bounded operators. On nonnegative interval [0, ∞), it has been shown that operator convex functions for bounded operators are also operator convex for unbounded operators [28].…”

Section: Jensen's Operator Inequalitymentioning

confidence: 99%

“…We present the definitions of von Neumann algebra and extended positive cone of von Neumann algebra as in [29,30]. For example, for bounded operators, f (x) = −x p is operator monotone for Theorem 2.6] and these functions are also operator monotone for unbounded operators with spectrum on I = (0, ∞) [31,28].…”

Section: Jensen's Operator Inequalitymentioning

confidence: 99%

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Numerical integration as a finite matrix approximation to multiplication operator

Sarmavuori

Särkkä

2019

Journal of Computational and Applied Mathematics

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We study the convergence of a family of numerical integration methods where the numerical integral is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, the convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen's operator inequality can be used to analyze the convergence of an improper numerical integral of a * Corresponding author function bounded by an operator convex function.

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Section: Definitionmentioning

confidence: 99%

Section: Jensen's Operator Inequalitymentioning

confidence: 99%

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Numerical integration as a finite matrix approximation to multiplication operator

Sarmavuori

Särkkä

2019

Journal of Computational and Applied Mathematics

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“…We do know that operator convex functions of positive bounded operators, like f (x) = x −1 , are operator convex for positive unbounded operators as well [48,Theorem 4.3].…”

Section: Lemma 27 Let a Be A Self-adjoint Operator With An Infinite M...mentioning

confidence: 99%

On the convergence of numerical integration as a finite matrix approximation to multiplication operator

Sarmavuori

Särkkä

2023

Calcolo

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We study the convergence of a family of numerical integration methods where the numerical integration is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for a wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen’s operator inequality can be used to analyse the convergence of an improper numerical integral of a function bounded by an operator convex function.

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Matrix Inequalities and Characterizations of Operator Monotone Functions

Dinh

Osaka

Tikhonov

2022

Trends in Mathematics

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Inequalities for the extended positive part of a von Neumann algebra related to operator-monotone and operator-convex functions

Cited by 5 publications

References 0 publications

Numerical integration as a finite matrix approximation to multiplication operator

Numerical integration as a finite matrix approximation to multiplication operator

On the convergence of numerical integration as a finite matrix approximation to multiplication operator

Matrix Inequalities and Characterizations of Operator Monotone Functions

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