Bases and Interpolation (Statement of the Problem)
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In this and the following two Lectures we will give a much deeper analysis of the complete truncated shift operators with a simple spectrum. In particular, we will give an exhausting answer to the question raised in the Introductory Lecture: when does a compression of the shift admit a spectral decomposition similar to the Spectral Theorem? The word “similar” can be interpreted literally, i.e. as “linearly similar”; we are thus seeking criteria for the model operator to be equivalent to a normal operator. It is not hard to see (cf. the “Supplements...” of this Lecture) that among the model operators with an inner characteristic function Θ and a resolvent meromorphic in the disk D the hypothesis of completeness and of the simplicity of the spectrum in fact does not diminish the generality — it is indeed a necessary condition for the operator TΘ to be similar to a normal operator. If one thus beforehand imposes this hypothesis (as we will do) the study of the corresponding spectral decomposition is reduced to the description of unconditional bases of eigenfunctions of the operators TΘ and T Θ * . We will precede our study of families of these eigenfunctions with some general results on bases in Hilbert space, the most important of which being the famous theorem of Bari. Then, proceeding to the eigenfunctions of the truncated shift, we will exhibit the connection between the basis problem and some classical interpolation problems in the spaces H2 and H∞ . This will also be the foundation for the main results of Lecture VI–VIII. The latter again will form the pattern upon which we solve the problem of spectrality (in the sense of Dunford) and of generalized spectrality for the model operator in Lecture IX–X.
KeywordsHilbert Space Interpolation Problem Riesz Basis Spectral Projection Unconditional Basis
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