Beurling quotient subspaces for covariant representations of product systems

Abstract

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.