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Iterative actions of normal operators. (English) Zbl 1361.46010

The authors study subsets of an infinite-dimensional Hilbert space \(\mathcal H\) of the form \[ \{A^ng:g\in \mathcal G, \, 0\leq n < L(g)\} \] where \(A\) is a bounded normal operator, \(\mathcal G\) is a finite or countably infinite subset of \(\mathcal H\) and \(L\) is a function: \(\mathcal G\to \{1,2,\dots\}\cup \{+\infty\}\). The questions asked are when this system is complete, a Bessel system, a basis, or a frame for \(\mathcal H\), and conversely, what can be said about the operator \(A\) if the system turns out to be of this kind. The motivation for this work comes from dynamic sampling theory and it is connected to several topics in functional and applied harmonic analysis.
In this thorough paper, a number of different results are obtained, many formulated in terms of the spectral measure of \(A\). Some of the results are negative, for example, one stating that if \(A\) is reductive and \(\mathcal G\) is finite, then the system \(\{A^ng\}_{g\in \mathcal G, 0\leq n < L(g)}\) cannot be a basis.

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
42C15 General harmonic expansions, frames
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