## Annals of Functional Analysis

### An operator inequality implying the usual and chaotic orders

#### Abstract

We prove that if positive invertible operators $A$ and $B$ satisfy an operator inequality $(B^{s/2}A^{(s-t)/2}B^tA^{(s-t)/2}B^{s/2})^{1\over{2s}}\geq B$ for some $t > s > 0$, then

(1) If $t \ge 3s-2 \ge 0$, then $\log B \geq \log A$, and if $t \ge s+2$ is additionally assumed, then $B \ge A$.

(2) If $s \in (0, 1/2)$, then $\log B \geq \log A$, and if $t \ge s+2$ is additionally assumed, then $B \ge A$.

It is an interesting application of the Furuta inequality. Furthermore we consider some related results.

#### Article information

Source
Ann. Funct. Anal., Volume 5, Number 1 (2014), 24-29.

Dates
First available in Project Euclid: 5 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1391614565

Digital Object Identifier
doi:10.15352/afa/1391614565

Mathematical Reviews number (MathSciNet)
MR3119108

Zentralblatt MATH identifier
06222696

#### Citation

Fujii, Jun Ichi; Fujii, Masatoshi; Nakamoto, Ritsuo. An operator inequality implying the usual and chaotic orders. Ann. Funct. Anal. 5 (2014), no. 1, 24--29. doi:10.15352/afa/1391614565. https://projecteuclid.org/euclid.afa/1391614565

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