Finite dimensional backward shift invariant subspaces of Arveson spaces
type: Item
instance of: scholarly article
title: Finite dimensional backward shift invariant subspaces of Arveson spaces
zbMATH Open document ID: 1019.46023
DOI: 10.1016/S0024-3795(02)00251-3
author: Vladimir Bolotnikov
author: Leiba Rodman
published in: Linear Algebra and its Applications
publication date: 2002-08-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 47B32
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 1790254
zbMATH Keywords: shift invariant subspaces
zbMATH Keywords: Arveson spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1790254