Existence of frames with prescribed norms and frame operator

Abstract

In this chapter we survey several recent results on the existence of frames with prescribed norms and frame operator. These results are equivalent to Schur-Horn type theorems which describe possible diagonals of positive self-adjoint operators with specified spectral properties. The first infinite dimensional result of this type is due to Kadison who characterized diagonals of orthogonal projections. Kadison’s theorem automatically gives a characterization of all possible sequences of norms of Parseval frames. We present some generalizations of Kadison’s result such as (a) the lower and upper frame bounds are specified, (b) the frame operator has two point spectrum, and (c) the frame operator has a finite spectrum.