## Banach Journal of Mathematical Analysis

### Comprehensive survey on an order preserving operator inequality

Takayuki Furuta

#### Abstract

In 1987, we established an operator inequality as follows; $A \ge B \ge 0$ $\Longrightarrow (A^{\frac {r}{2}} A^p A^{\frac {r}{2}})^{\frac{1}{q}} \ge (A^{\frac {r}{2}} B^p A^{\frac {r}{2}})^{\frac{1}{q}}$ holds for (*) $p \ge 0$, $q \ge 1$, $r \ge 0$ with $(1+r)q \ge p+r.$ It is an extension of Löwner-Heinz inequality. The purpose of this paper is to explain geometrical background of the domain by (*), and to give brief survey of recent results of its applications.

#### Article information

Source
Banach J. Math. Anal., Volume 7, Number 1 (2013), 14-40 .

Dates
First available in Project Euclid: 22 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1358864546

Digital Object Identifier
doi:10.15352/bjma/1358864546

Mathematical Reviews number (MathSciNet)
MR3004264

Zentralblatt MATH identifier
1276.47020

Subjects
Primary: 47A63
Secondary: 47B20 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.) 47H05

#### Citation

Furuta, Takayuki. Comprehensive survey on an order preserving operator inequality. Banach J. Math. Anal. 7 (2013), no. 1, 14--40. doi:10.15352/bjma/1358864546. https://projecteuclid.org/euclid.bjma/1358864546

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