Ellipsoidal tight frames and projection decompositions of operators

Abstract

We prove the existence of tight frames whose elements lie on an arbitrary ellipsoidal surface within a real or complex separable Hilbert space $\mathcal{H}\, $, and we analyze the set of attainable frame bounds. In the case where $\mathcal{H}\,$ is real and has finite dimension, we give an algorithmic proof. Our main tool in the infinite dimensional case is a result we have proven which concerns the decomposition of a positive invertible operator into a strongly converging sum of (not necessarily mutually orthogonal) self-adjoint projections. This decomposition result implies the existence of tight frames in the ellipsoidal surface determined by the positive operator. In the real or complex finite dimensional case, this provides an alternate (but not algorithmic) proof that every such surface contains tight frames with every prescribed length at least as large as $\dim\mathcal{H}\, $. A corollary in both finite and infinite dimensions is that every positive invertible operator is the frame operator for a spherical frame.