Beurling quotient subspaces for covariant representations of product systems

Let \((\sigma , V^{(1)}, \dots , V^{(k)})\) be a pure doubly commuting isometric representation of the product system \({\mathbb {E}}\) on a Hilbert space \({\mathcal {H}}_{V}.\) A \(\sigma \)-invariant subspace \({\mathcal {K}}\) is said to be Beurling quotient subspace of \({\mathcal {H}}_{V}\) if there exist a Hilbert space \({\mathcal {H}}_W,\) a pure doubly commuting isometric representation \((\pi , W^{(1)}, \dots , W^{(k)})\) of \({\mathbb {E}}\) on \({\mathcal {H}}_W\) and an isometric multi-analytic operator \(M_\Theta :{{\mathcal {H}}_W} \rightarrow {\mathcal {H}}_{V}\), such that

$$\begin{aligned} {\mathcal {K}}={\mathcal {H}}_{V}\ominus M_{\Theta }{\mathcal {H}}_W, \end{aligned}$$

where \(\Theta : {\mathcal {W}}_{{\mathcal {H}}_W} \rightarrow {\mathcal {H}}_{V} \) is an inner operator and \({\mathcal {W}}_{{\mathcal {H}}_W}\) is the generating wandering subspace for \((\pi , W^{(1)}, \dots , W^{(k)}).\) In this article, we prove the following characterization of the Beurling quotient subspaces: A subspace \({\mathcal {K}}\) of \({\mathcal {H}}_{V}\) is a Beurling quotient subspace if and only if

$$\begin{aligned}&(I_{E_{j}}\otimes ( (I_{E_{i}}\otimes P_{{\mathcal {K}}}) - \widetilde{T}^{(i) *}\widetilde{T}^{(i)}))(t_{i,j} \otimes I_{{\mathcal {H}}_{V}})\\&(I_{E_{i}}\otimes ( (I_{E_{j}}\otimes P_{{\mathcal {K}}})- \widetilde{T}^{(j) *}\widetilde{T}^{(j)}))=0, \end{aligned}$$

where \(\widetilde{T}^{(i)}:=P_{{\mathcal {K}}}\widetilde{V}^{(i)} (I_{E_{i}} \otimes P_{{\mathcal {K}}})\) and \( 1 \le i,j\le k.\) As a consequence, we derive a concrete regular dilation theorem for a pure, completely contractive covariant representation \((\sigma , V^{(1)}, \dots , V^{(k)})\) of \({\mathbb {E}}\) on a Hilbert space \({\mathcal {H}}_{V}\) 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.