Logarithmic order and dual logarithmic order

Abstract

We shall define the following four orders for strictly positive operators A and B on a Hilbert space H. Strictly logarithmic order (denoted by A≻ sl B) is defined by $\frac{{A - I}}{{\log A}} > \frac{{B - I}}{{\log B}}$. Logarithmic order (denoted by A ≻ l B) is defined by $\frac{{A - I}}{{\log A}} \geqslant \frac{{B - I}}{{\log B}}$. Strictly dual logarithmic order (denoted by A ≻ _sdl B) is defined by $\frac{{A\log A}}{{A - I}} > \frac{{B\log B}}{{B - I}}$ Dual Logarithmic order (denoted by A≻ dl B) is defined by $\frac{{A\log A}}{{A - I}} \geqslant \frac{{B\log B}}{{B - I}}$