Generalized frame operator, lower semi-frames and sequences of translates

Abstract

Given a sequence of elements ξ = {ξ n } n∈N of a Hilbert space, an operator T ξ is defined as the operator associated to a sesquilinear form determined by ξ. This operator is in general different to the classical frame operator but possesses some remarkable properties. For instance, T ξ is self-adjoint (in an specific space), unconditionally defined and, when ξ is a lower semi-frame, T ξ gives a simple expression of a dual of ξ. The operator T ξ and lower semi-frames are studied in the context of sequences of integer translates.

Lower semi-frames and metric operators

“…If D(C φ ) is dense, then of course H φ = H (i.e., T φ is a non-negative self-adjoint operator on H) and S φ ⊂ T φ . In [22], it was proved that in the discrete setting (X = N) we may have a strict inclusion S φ T φ (recall that in our paper S φ is weakly defined, therefore in the discrete setting it corresponds to the operator W φ in [22]).…”

“…The motivation behind this name is that when φ is a continuous frame then T φ = S φ . The generalized frame operator has been studied in [21,22] in the discrete case and a preliminary extension to the continuous setting has been given in [18].…”

“…It turns out that most results from [21,22] can be extended to the continuous case and that is one of the aims of this present paper. The results are reported in Sect.3, but we give here a brief summary.…”

“…In recent papers, one of us (RC) [21,22] has analyzed sesquilinear forms defined by sequences in Hilbert spaces and operators associated to them by means of representation theorems. In particular, he derived results about lower semi-frames and duality.…”

“…In this paper we do not confine ourselves to extend results of [21,22], but actually we set two more goals. From one hand, given a lower semi-frame φ : X → H with D(C φ ) dense in H, we consider general powers T −k φ φ with k ≥ 0.…”