# When is an inner derivation a Fredholm operator?

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 ...).

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