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 H , and we analyze the set of attainable frame bounds. In the case where 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 H . A corollary in both finite and infinite dimensions is that every positive invertible operator is the frame operator for a spherical frame.

Projection decomposition in multiplier algebras

“…In this article we generalize the main result of [5] to certain multiplier algebras, stated as follws. (iv) A is the sum of finitely many projections belonging to A.…”

“…They proved that a sufficient condition for a positive bounded operator A ∈ B(H) to be a (possibly infinite) sum of projections converging in the strong operator topology is that its essential norm A ess is >1 (see [5], Theorem 2). This result served as a basis for further work by Kornelson and Larson [13] and then by Antezana et al [1] on decompositions of positive operators into strongly converging sums of rank one positive operators with preassigned norms.…”

“…Acknowledgments V. Kaftal [5,13] that stimulated this project. V. Kaftal and S. Zhang were partially supported by grants from the Charles Phelps Taft Research Center.…”

Strong sums of projections in von Neumann factors

“…For infinite sums with convergence in the strong operator topology, this question arose naturally from work on ellipsoidal tight frames by Dykema, Freeman, Kornelson, Larson, Ordower, and Weber in [5]. They proved that a sufficient condition for a positive bounded operator A ∈ B(H ) + to be the sum of projections is that its essential norm A e is larger than one [5,Theorem 2].…”

Finite Frame Varieties: Nonsingular Points, Tangent Spaces, and Explicit Local Parameterizations

“…For structured frames, many existence results and construction techniques have been demonstrated. The existence of unit-norm and ellipsoidal tight frames was considered in [9,14,19], the existence of (μ, S)-frames was shown in [10,19], and the results on the existence of equiangular tight frames has been demonstrated in [5,26,28]. These existence proofs are all constructive, but numerically efficient constructions have also been considered [12,25,29] and the frame potential introduced by Benedetto and Fickus [1] provides a robust iterative method for constructing finite unit-norm tight frames.…”