This question is related to Derivation into dense ideal of Banach algebras. Depending on which way it goes, an answer to one might answer the other (this is elaborated below).
Let $H$ be a Hilbert space, $K(H)$ the algebra of compact operators on $H$, and $F(H)$ the ideal in $K(H)$ of finite rank operators. Suppose that $D:K(H)\to F(H)$ is a bounded derivation, meaning $D$ is a bounded linear map such that $D(ab)=aD(b)+D(a)b$ for all $a,b\in K(H)$.
Question: Must $D$ be an inner derivation, in the sense that there exists $x\in F(H)$ such that $D(a)=ax-xa$ for all $a\in K(H)$?
Some context including a reformulation of the question:
- In the question linked above, there is a dense ideal $I$ in a Banach algebra $A$, the hypothesis is that every bounded derivation $A\to I$ is inner, and the question is whether this implies every bounded derivation $A\to A$ is inner. If the answer to that question is yes, then the answer to this one is no, because $F(H)$ is a dense ideal in $K(H)$ and $K(H)$ has outer derivations, e.g. $a\mapsto aS-Sa$ where $S$ is a shift operator. Hence, if the answer to this one is yes, the answer to that one is no. But a no answer on either wouldn't necessarily help answer the other.
- Every bounded derivation $K(H)\to K(H)$ has the form $a\mapsto ax-xa$ for some $x\in B(H)$ (proof appears in Hanno's answer at the linked question), and every $x\in B(H)$ defines such a derivation on $K(H)$. These can be outer as indicated above. But here the much stricter requirement is that the range is in $F(H)$, and the question can be reformulated as:
Question, Version 2: If $x$ is in $B(H)$ and $ax-xa$ has finite rank for all $a\in K(H)$, must $x$ be scalar plus finite rank?
