2

Consider the following definition:

Let $\mathcal T$ be the family of bounded operators $T$ on the separable infinite dimensional complex Hilbert space $\mathfrak H$ with the following property: $T\in\mathcal T$ if and only if there exists $c\in\mathbb C$ such that, for every orthonormal basis $\{\xi_n\}_n$, $\sum_n(\xi_n,T\xi_n)=c$ (namely convergence for every o.n.b. and same sum for every o.n.b.)

Clearly, $\mathcal T\supset \mathcal I_1$, the trace class ideal. Is it actually larger? I vaguely remember that they coincide, but I could not find any reference, neither did I manage to prove or disprove it...

Gherardo
  • 41
  • Clearly $\mathcal T$ and $\mathcal I_1$ share the positive operators, and thus also the normal operators which can be written as a linear combination $T=\sum_{n=0}^3i^nT_n$ of four pairwise commuting positive operators such that $|T|^2=\sum_{n=0}^3T_n^2$ and $|T|\leqslant\sum_{n=0}^3T_n$. But what about non normal operators? – Gherardo Aug 27 '16 at 10:20
  • This question might be useful https://math.stackexchange.com/questions/208508/a-characterization-of-trace-class-operators – hyperkahler Dec 22 '18 at 17:01

0 Answers0