Norms of multiplication operators: answering Fialkow--Loebl question

Abstract

Let $\mathcal{M}$ be a factor equipped with a semi-finite faithful normal trace $\tau$. Let $E(0,\infty )$ be a symmetrically normed function space and $E(\mathcal{M},\tau )$ be the corresponding symmetrically normed operator space. Suppose that $a,b$ are $\tau$-measurable operators affiliated with $\mathcal{M}$. It is shown that the range of the multiplication operator $S_{a,b}:x\mapsto axb$ on $\mathcal{M}$ is contained in $E(\mathcal{M},\tau )$ if and only if $\mu (a)\mu (b)$ belongs to $E(0,\infty )$, where $\mu (x)$ stands for the generalized singular value function of a $\tau$-measurable operators $x$ affiliated with $\mathcal{M}$. Moreover, we have $$\Vert S_{a,b}{\Vert }_{\mathcal{M}\to E(\mathcal{M},\tau )}=\Vert \mu (a)\mu (b){\Vert }_{E(0,\infty )},$$ which answers a question by Fialkow and Loebl (1984). We also consider the quasi-normed case, and show that the natural quasi-norm of weak $L_p$-space, $0<p<\infty$, is not monotone with respect to the logarithmic submajorisation.