Derivations on symmetric quasi-Banach ideals of compact operators

Abstract

Let $\mathcal{I,J}$ be symmetric quasi-Banach ideals of compact operators on an infinite-dimensional complex Hilbert space $H$, let $\mathcal{J:I}$ be a space of multipliers from $\mathcal{I}$ to $\mathcal{J}$. Obviously, ideals $\mathcal{I}$ and $\mathcal{J}$ are quasi-Banach algebras and it is clear that ideal $\mathcal{J}$ is a bimodule for $\mathcal{I}$. We study the set of all derivations from $\mathcal{I}$ into $\mathcal{J}$. We show that any such derivation is automatically continuous and there exists an operator $a\in\mathcal{J:I}$ such that $\delta(\cdot)=[a,\cdot]$, moreover $\|a\|_{\mathcal{B}(H)}\leq\|\delta\|_\mathcal{I\to J}\leq 2C\|a\|_\mathcal{J:I}$, where $C$ is the modulus of concavity of the quasi-norm $\|\cdot\|_\mathcal{J}$. In the special case, when $\mathcal{I=J=K}(H)$ is a symmetric Banach ideal of compact operators on $H$ our result yields the classical fact that any derivation $\delta$ on $\mathcal{K}(H)$ may be written as $\delta(\cdot)=[a,\cdot]$, where $a$ is some bounded operator on $H$ and $\|a\|_{\mathcal{B}(H)}\leq\|\delta\|_\mathcal{I\to I}\leq 2\|a\|_{\mathcal{B}(H)}$.