$\mu $-Norm of an Operator

Let $(\mathcal X,\mu )$ be a measure space. For any measurable set $Y\subset \mathcal X$ let $\mathbf 1_Y: \mathcal X\to \mathbb{R} $ be the indicator of $Y$ and let $\pi _Y^{}$ be the orthogonal projection $L^2(\mathcal X)\ni f\mapsto {\pi _Y^{}}_{} f = \mathbf 1_Y f$. For any bounded operator $W$ on $L^2(\mathcal X,\mu )$ we define its $\mu $-norm $\|W\|_\mu = \inf _\chi \sqrt {\sum \mu (Y_j)\|W\pi _Y^{}\|^2}$, where the infimum 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 constructing a quantum entropy.