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 $(langle Tu_j,u_nrangle)_{n,j=1}^infty$ induced by the set of all orthonormal bases $(u_n)$ in $H$. We discuss the following problem:

Problem: Let $Bsubset{bf N}times{bf N}$ 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
$$
langle Tu_j,u_nrangle=a_{nj}qquad(n,j)in B?
$$

This is a joint work with Yu. Tomilov.

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