Relations among operator orders and operator inequalities (Recent Topics on Operator inequalities)

The Furuta inequality and the chaotic orderIn what follows, an operator means a bounded linear operator on a Hilbert space $H$ and is denoted by a capital letter.An operator $T$ is said to be positive (denoted by $T\geq 0$ ) if $(Tx, x)\geq 0$ for all $x\in H,$ and also $T$ is said to be strictly positive (denoted by $T>0$) if $T$ is positive and invertible.We start this report with introduction of the following order preserving operator inequalities. Theorem $\mathrm{F}$ (Furuta inequality [5]).