$μ$-norm of an operator

Let $({\cal X},μ)$ be a measure space. For any measurable set $Y\subset{\cal X}$ let $1_Y : {\cal X}\to{\mathbb R}$ be the indicator of $Y$ and let $π_Y$ be the orthogonal projector $L^2({\cal X})\ni f\mapstoπ_Y f = 1_Y f$. For any bounded operator $W$ on $L^2({\cal X},μ)$ we define its $μ$-norm $\|W\|_μ= \inf_χ\sqrt{\sum μ(Y_j) \|Wπ_Y\|^2}$, where the infinum is taken over all measurable partitions $χ= \{Y_1,\ldots,Y_J\}$ of ${\cal X}$. We present some properties of the $μ$-norm and some computations. Our main motivation is the problem of the construction of a quantum entropy.

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