Traces of compact operators (Q2266216)

Let H be a separable Hilbert space, T a compact bounded linear operator in H, \(\phi =\{\phi_ j\}^{\infty}_{j=1}\) an orthonormal basis of H. Then \(\phi \in Dom\{tr T\}\) if the series \(\sum^{\infty}_{j=1}<T\phi_ j,\phi_ j>\) converges, and if so, the sum is denoted by \(tr_{\phi}T\). The image of tr T is denoted \(R\{tr T\}.\) Under the above conditions, the following is proven: If \(Dom\{tr T\}\neq \emptyset,\) then \(R\{\) tr \(T\}\) is either a point or a straight line in the complex plane or the whole complex plane. Related results are given.