Matrices of operators

Let $T$ be a linear operator on a separable infinite-dimensional Hilbert space $H$. Then $T$ allows for a variety of matrix representations $(\langle Tu_j,u_n\rangle)_{n,j=1}^\infty$ induced by the set of all orthonormal bases $(u_n)$ in $H$. We discuss the following problem:

Problem: Let $B\subset{\bf N}\times{\bf N}$ be a subset and $a_{nj}\quad(j,n)\in B$ given complex numbers. What are natural assumptions on $B$ and $a_{nj}$ to ensure that there exists an orthonormal basis $(u_n)$ such that
$$
\langle Tu_j,u_n\rangle=a_{nj}\qquad(n,j)\in B?
$$

This is a joint work with Yu. Tomilov.

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