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5 - Beurling–Lax Theorems Based on Contractive Multipliers

Published online by Cambridge University Press:  09 December 2021

Joseph A. Ball
Affiliation:
Virginia Tech
Vladimir Bolotnikov
Affiliation:
College of William and Mary, Virginia
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Summary

Chapter 5 obtains Beurling–Lax-type representation theorems for isometrically included forward-shift-invariant subspaces as the range of a partially isometric (i.e., McCT-inner) multiplier coming off a (non-weighted) Hardy–Fock space. More generally, a contractively included forward-shift-invariant subspace of a weighted Hardy–Fock space is characterized as the range of a contractive multiplier coming off a (non-weighted) Hardy–Fock space with intrinsic norm equal to the lifted norm induced by the representation as the range of the associated contractive multiplier. When some additional conditions are satisfied, it is possible to obtain more explicit transfer-function realizations for the Beurling–Lax representer. These additional conditions are intimately connected with the question as to when the Brangesian complement of a forward/backward shift-invariant subspace is backward/forward shift-invariant. An example is given to show that, unlike as in the classical setting or in the case where the shift-invariant subspace is isometrically included in the ambient weighted Hardy–Fock space, it can happen that the question has a negative answer.

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Publisher: Cambridge University Press
Print publication year: 2021

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