Beurling quotient subspaces for covariant representations of product systems

Article Type

Research Article

Publication Title

Annals of Functional Analysis

Abstract

Let (σ, V(1), ⋯ , V(k)) be a pure doubly commuting isometric representation of the product system E on a Hilbert space HV. A σ -invariant subspace K is said to be Beurling quotient subspace of HV if there exist a Hilbert space HW, a pure doubly commuting isometric representation (π, W(1), ⋯ , W(k)) of E on HW and an isometric multi-analytic operator MΘ: HW→ HV , such that K=HV⊖MΘHW, where Θ:WHW→HV is an inner operator and WHW is the generating wandering subspace for (π, W(1), ⋯ , W(k)). In this article, we prove the following characterization of the Beurling quotient subspaces: A subspace K of HV is a Beurling quotient subspace if and only if (IEj⊗((IEi⊗PK)-T~(i)∗T~(i)))(ti,j⊗IHV)(IEi⊗((IEj⊗PK)-T~(j)∗T~(j)))=0, where T~(i):=PKV~(i)(IEi⊗PK) and 1 ≤ i, j≤ k. As a consequence, we derive a concrete regular dilation theorem for a pure, completely contractive covariant representation (σ, V(1), ⋯ , V(k)) of E on a Hilbert space HV which satisfies Brehmer–Solel condition and using it and the above characterization, we provide a necessary and sufficient condition that when a completely contractive covariant representation is unitarily equivalent to the compression of the induced representation on the Beurling quotient subspace. Further, we study the relation between Sz. Nagy–Foias-type factorization of isometric multi-analytic operators and joint invariant subspaces of the compression of the induced representation on the Beurling quotient subspace.

DOI

https://10.1007/s43034-023-00301-0

Publication Date

10-1-2023

Comments

Open Access, Green

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