Two Inner Sequences Based Invariant Subspaces in $${{H}^{2} (\mathbb{D}^{2})}$$ H 2 ( D 2 )

Yixin Yang

doi:10.1007/s00020-013-2083-z

Abstract

Let M be a shift invariant subspace in the vector-valued Hardy space ${H_{E}^{2}(\mathbb{D})}$. The Beurling–Lax–Halmos theorem says that M can be completely characterized by ${\mathcal{B}(E)}$-valued inner function ${\Theta}$. When ${E = H^{2}(\mathbb{D}),\,H_{E}^{2}(\mathbb{D})}$ is the Hardy space on the bidisk ${H^{2}(\mathbb{D}^2)}$. Recently, Qin and Yang (Proc Am Math Soc, 2013) determines the operator valued inner function ${\Theta(z)}$ for two well-known invariant subspaces in ${H^{2}(\mathbb{D}^{2})}$. This paper generalizes the ${\Theta(z)}$ by Qin and Yang (Proc Am Math Soc, 2013) and deal with the structure of ${M = {\Theta}(z)H^{2}(\mathbb{D}^{2})}$ when M is an invariant subspace in ${H^{2}(\mathbb{D}^{2})}$. Unitary equivalence, spectrum of the compression operator and core operator are studied in this paper.

Highlights

Let D be the unit disk with boundary T in the complex plane C
Beurling–Lax–Halmos theorem says that such M is of the form M = Θ(z)HE2 (D) for some B(E)-valued inner function Θ
By the Beurling–Lax–Halmos theorem, the Tz-invariant subspace M is of the form M = Θ(z)H2(D2) for some B(H2(w))-valued inner function Θ

Summary

Introduction

Let D be the unit disk with boundary T in the complex plane C. Let HE2 (D) be the E-valued Hardy space on the unit disk. M is said to be an invariant subspace if it is invariant both under the action of Tz and Tw. Let N = H2(D2) M be the corresponding quotient space. By the Beurling–Lax–Halmos theorem, the Tz-invariant subspace M is of the form M = Θ(z)H2(D2) for some B(H2(w))-valued inner function Θ. For an invariant subspace M , there is a B(H2(z))-valued inner function Φ such that M = Φ(w)H2(D2). All the information of M are encoded in the two operator-valued inner functions Θ and Φ. [11] determined these two functions for two well-known invariant subspaces [8, 12, 13] in H2(D2) and they turn out to be strikingly simple. We will focus on the invariant subspace given by the following operatorvalued inner function.

Structure of M

Unitary Equivalence

Spectrum

Core Operator