Example of a "weak" Compact operator

Question

Let $\mathcal{H}$ be a separable Hilbert space with ONB $\{e_n\}_n$. Following my previous question, I would like to find an example of a bounded operator $A$ such that

• $A$ is compact.
• $A$ is not $p$-Schatten for any $p\geq 1$.
• $A$ is not finite rank, diagonal, Toeplitz or Hankel.

Answer 1

Take any sequence $\{x_n\}_n$ in $c_0$ which is not in $l^p$ for any $p$, and consider the corresponding diagonal operator with diagonal entries $x_n$. This operator has all of the required properties, except that it is diagonal.

To make it non-diagonal, conjugate it by a random unitary matrix and, unless you are very very unlucky, it will not be diagonal, Toeplitz or Hankel, and hence will satisfy all of the conditions.