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Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces. (English) Zbl 0821.46008
Summary: A problem of enduring interest in connection with the study of frames in Hilbert 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 \(\{x_ n\}^ \infty_{n= 1}\) leaves a Riesz basis if and only if the frame is Besselian (i.e., \(\sum^ \infty_{n= 1} a_ n x_ n\) converges \(\Leftrightarrow (a_ n)\in \ell^ 2\)).

MSC:
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
47A53 (Semi-) Fredholm operators; index theories
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