Let $\mathcal{B}(H)$ denote the algebra of bounded operators on a Hilbert space $H$. I'm interested in inner derivations acting on the Schatten ideals $L^p\subseteq\mathcal{B}(H)$ (defined by operators $A\in\mathcal{B}(H)$ such that $\operatorname{tr}|A|^p<\infty$). More precisely, given $D\in\mathcal{B}(H)$ one constructs $\delta$ as an inner derivation on the Banach space $L^p$ by setting $\delta(A)=[A,D]$. When is $\delta$ a Fredholm operator (as an operator on $L^p$)? That is, when are the kernel and the cokernel finite dimensional?

Since $\delta$ is a derivation, it follows that if $\delta(A)=0$, then $\delta(p(A))=0$ for every polynomial $p$. Hence, if there are non-trivial elements in the kernel, it is likely not finite-dimensional (well, $A$ could of course be nilpotent, or proportional to the identity, or idempotent ...).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.