Finite dimensional backward shift invariant subspaces of Arveson spaces (Q1611871)

An \(n\)-dimensional space is an invariant subspace of the backward shift operator \(R(F) = \frac{F(z) - F(0)}{z}\) defined on the Hilbert space of \(\mathbb C^p\)-valued functions, \(\mathbf H ^p_2\), if and only if it is spanned by the columns of a \(p\times n\) matrix valued function of the form \(F(z) = C(I_n - zA)^{-1}\) where \(C \in \mathbb C^{p\times n}\), \(A \in \mathbb C^{n\times n}\) are such that \(\sigma (A)\) is inside the unit disk and \(\bigcap_{j=0}^{\infty} \text{Ker}(CA^j) = \{ 0 \}\). The authors extend such known results to give a description of finite dimensional backward shift invariant subspaces where \(\mathbf H ^p_2\) is replaced by a multivariable analogue known as Arveson spaces. The one variable Arveson space is the reproducing kernel Hilbert space \({\mathcal H} (K)\) with kernel \(K(z,w) = \frac{1}{1-\langle z, w \rangle}\). The authors extend this to spaces \({\mathcal H}^p (K)\) of \(\mathbb C^p\)-valued vector functions with entries in \({\mathcal H} (K)\) as well.