On zero-diagonal operators and traces

Abstract

A Hilbert space operator A A is called zero-diagonal if there exists an orthonormal basis ϕ = { e j } j = 1 ∞ \phi = \{ {e_j}\} _{j = 1}^\infty such that ⟨ A e j , e j ⟩ = 0 \left \langle {A{e_j},{e_j}} \right \rangle = 0 for all j j . It is known that T T is the norm limit of a sequence { A k } \{ {A_k}\} of zero-diagonal operators iff 0 ∈ W e ( T ) 0 \in {W_e}(T) , the essential numerical range of T T . Our first result says that if 0 ∈ W e ( T ) 0 \in {W_e}(T) and J \mathcal {J} is an ideal of compact operators strictly larger than the trace class, then the sequence { A k } \{ {A_k}\} can be chosen so that | T − A k | J → 0 |T - {A_k}{|_\mathcal {J}} \to 0 ( J \mathcal {J} cannot be replaced by the trace class!). If A A is zero-diagonal, then the series ∑ j = 1 ∞ ⟨ A e j , e j ⟩ \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 A with respect to the basis ϕ \phi ". Our second result provides, for each operator T T , the structure of the set of all possible "traces" of T 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.