Derivations on symmetric quasi-Banach ideals of compact operators

Abstract

Let $\mathcal{I},\mathcal{J}$ be symmetric quasi-Banach ideals of compact operators on an infinite-dimensional complex Hilbert space $H$, let $\mathcal{J}:\mathcal{I}$ 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 $a\in \mathcal{J}:\mathcal{I}$ such that $\delta (\cdot )=[a,\cdot ]$, moreover $\Vert a{\Vert }_{\mathcal{B}(H)}\le \Vert \delta {\Vert }_{\mathcal{I}\to \mathcal{J}}\le 2C\Vert a{\Vert }_{\mathcal{J}:\mathcal{I}}$, where $C$ is the modulus of concavity of the quasi-norm $\Vert \cdot {\Vert }_{\mathcal{J}}$. In the special case, when $\mathcal{I}=\mathcal{J}=\mathcal{K}(H)$ 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 $\Vert a{\Vert }_{\mathcal{B}(H)}\le \Vert \delta {\Vert }_{\mathcal{I}\to \mathcal{I}}\le 2\Vert a{\Vert }_{\mathcal{B}(H)}$.

21 pages

References in corpus (1)

• Derivations in the Banach ideals of $\tau$-compact operators