Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces

Holub, James R.

doi:10.1090/S0002-9939-1994-1204376-4

Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces
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Proc. Amer. Math. Soc. 122 (1994), 779-785 Request permission

Abstract:

A problem of enduring interest in connection with the study of frames in Hubert space is that of characterizing those frames which can essentially be regarded as Riesz bases for computational purposes or which have certain desirable properties of Riesz bases. In this paper we study several aspects of this problem using the notion of a pre-frame operator and a model theory for frames derived from this notion. In particular, we show that the deletion of a finite set of vectors from a frame $\{ {x_n}\} _{n = 1}^\infty$ leaves a Riesz basis if and only if the frame is Besselian (i.e., ${\sum } _{n = 1}^\infty {a_n}{x_n}$ converges $\Leftrightarrow ({a_n}) \in {l^2}$).

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