On Zero-Diagonal Operators and Traces
Peng Fan, C. K. Fong, Domingo Antonio Herrero · Proceedings of the American Mathematical Society · 1987
A Hilbert space operator $A$ is called zero-diagonal if there exists an orthonormal basis $\phi = \{ {e_j}\} _{j = 1}^\infty$ such that $\left \langle {A{e_j},{e_j}} \right \rangle = 0$ for all $j$. It is known that $T$ is the norm limit of a sequence $\{ {A_k}\}$ of zero-diagonal operators iff $0 \in {W_e}(T)$, the essential numerical range of $T$. Our first result says that if $0 \in {W_e}(T)$ and $\mathcal {J}$ is an ideal of compact operators strictly larger than the trace class, then the sequence $\{ {A_k}\}$ can be chosen so that $|T - {A_k}{|_\mathcal {J}} \to 0$ ($\mathcal {J}$ cannot be replaced by the trace class!). If $A$ is zero-diagonal, then the series $\sum _{j = 1}^\infty \left \langle {A{e_j},{e_j}} \right \rangle$ converges to a value (zero) that can be called "the trace of $A$ with respect to the basis $\phi$". Our second result provides, for each operator $T$, the structure of the set of all possible "traces" of $T$ (in the above sense). In particular, this set is always either the whole complex plane, a straight line, a singleton, or the empty set.