Relations between two inequalities \((B^{\frac r2} A^p B^{\frac r2})^{\frac r{p+r}}\geq B^r\) and \(A^p\geq(A^{\frac p2} B^r A^{\frac p2})^{\frac p{p+r}}\) and their applications
type: Item
instance of: scholarly article
title: Relations between two inequalities \((B^{\frac r2} A^p B^{\frac r2})^{\frac r{p+r}}\geq B^r\) and \(A^p\geq(A^{\frac p2} B^r A^{\frac p2})^{\frac p{p+r}}\) and their applications
zbMATH Open document ID: 1028.47013
DOI: 10.1007/BF01193670
author: Masatoshi Ito
author: Takeaki Yamazaki
published in: Integral Equations and Operator Theory
publication date: 2003-07-15T00:00:00Z
review text: Let \(A\) and \(B\) be positive bounded linear operators on a complex Hilbert space and let \(p\) and \(r\) be non-negative real numbers. In this paper, the authors give relations between the two inequalities \((B^{r/2}A^p B^{r/2})^{r/p+r}\geq B^r\) and \(A^p\geq(A^{p/2} B^rA^{p/2})^{p/p+r}\) as follows: (i) If \((B^{r/2}A^pB^{r/2})^{r/p+r}\geq B^r\), then \(A^p\geq(A^{p/2} B^rA^{p /2})^{p/p+r}\). (ii) If \(A^p\geq(A^{p/2}B^rA^{r/2})^{p/p+r}\) and \(N(A) \subset N(B)\), then \((B^{r/2}A^pB)^{r/p+r}\geq B^r\), where \(N(X)\) denotes the null space of the operator \(X\).
reviewed by: Omar Hirzallah
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 47B20
zbMATH DE Number: 1890561
zbMATH Keywords: operator inequalities
zbMATH Keywords: positive bounded linear operator
zbMATH Keywords: complex Hilbert space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1890561