$\mu$-norm of an operator

Abstract

Let $(\mathcal{X},\mu )$ be a measure space. For any measurable set $Y\subset \mathcal{X}$ let $1_Y:\mathcal{X}\to \mathbb{R}$ be the indicator of $Y$ and let ${\pi }_Y$ be the orthogonal projector $L^2(\mathcal{X})\ni f\mapsto {\pi }_{Y}f=1_{Y}f$. For any bounded operator $W$ on $L^2(\mathcal{X},\mu )$ we define its $\mu$-norm $\Vert W{\Vert }_{\mu }={\inf }_{\chi }\sqrt{\sum \mu (Y_j)\Vert W{\pi }_Y{\Vert }^2}$, where the infinum is taken over all measurable partitions $\chi =\{Y_1,\dots ,Y_J\}$ of $\mathcal{X}$. We present some properties of the $\mu$-norm and some computations. Our main motivation is the problem of the construction of a quantum entropy.

References in corpus (1)

• Entropy rate of message sources driven by quantum walks

Cited by in corpus (1)

• $\mu$-norm and regularity