  Given a bounded linear operator $T$ on separable Hilbert space, we develop an
approach allowing one to construct a matrix representation for $T$ having
certain specified algebraic or asymptotic structure. We obtain matrix
representations for $T$ with preassigned bands of the main diagonals, with an
upper bound for all of the matrix elements, and with entrywise polynomial lower
and upper bounds for these elements. In particular, we substantially generalize
and complement our results on diagonals of operators from [46] and other
related results. Moreover, we obtain a vast generalization of a theorem by
Stout (1981), and (partially) answer his open question. Several of our results
have no analogues in the literature.
