Two Combinations of Unitary Operators and Frame Representations

摘要

In this paper, we prove that the norm closure of all linear combinations of two unitary operators is equal to the norm closure of all invertible operators in B(H). We apply the results to frame representations and give some simple and alternative proofs of the propositions in “P. G. Casazza, Every frame is a sum of three (but not two) orthonormal bases-and other frame representations, J. Fourier Anal. Appl., 4(6)(1998), 727-732.”