Invariant subspaces of the Dirichlet space

Bergman Spaces on Disconnected Domains

Canadian Journal of Mathematics ◽

10.4153/cjm-1996-011-5 ◽

1996 ◽

Vol 48 (2) ◽

pp. 225-243

Author(s):

Alexandru Aleman ◽

Stefan Richter ◽

William T. Ross

Keyword(s):

Measure Zero ◽

Bergman Spaces ◽

Dirichlet Space ◽

Invariant Subspaces ◽

Compact Set ◽

Bounded Region ◽

Area Measure ◽

Weak Star ◽

Harmonic Dirichlet Space ◽

Disconnected Domains

AbstractFor a bounded region G ⊂ ℂ and a compact set K ⊂ G, with area measure zero, we will characterize the invariant subspaces ℳ (under ƒ → zƒ) of the Bergman space (G \ K), 1 ≤ p < ∞, which contain (G) and with dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K. When G \ K is connected, we will see that dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K and thus in this case we will have a complete description of the invariant subspaces lying between (G) and (G \ K). When p = ∞, we will remark on the structure of the weak-star closed z-invariant subspaces between H∞(G) and H∞(G \ K). When G \ K is not connected, we will show that in general the invariant subspaces between (G) and (G \ K) are fantastically complicated. As an application of these results, we will remark on the complexity of the invariant subspaces (under ƒ → ζƒ) of certain Besov spaces on K. In particular, we shall see that in the harmonic Dirichlet space , there are invariant subspaces ℱ such that the dimension of ζℱ in ℱ is infinite.

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Invariant subspaces of the Dirichlet space and commutative algebra

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crll.2004.025 ◽

2004 ◽

Vol 2004 (569) ◽

Author(s):

Xiang Fang

Keyword(s):

Commutative Algebra ◽

Dirichlet Space ◽

Invariant Subspaces

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Invariant subspaces of parabolic self-maps in the Dirichlet space

Journal of Functional Analysis ◽

10.1016/j.jfa.2014.02.003 ◽

2014 ◽

Vol 266 (7) ◽

pp. 4115-4120

Author(s):

Alfonso Montes-Rodríguez ◽

Manuel Ponce-Escudero

Keyword(s):

Dirichlet Space ◽

Invariant Subspaces

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Invariant subspaces of the harmonic Dirichlet space with large co-dimension

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-96-03243-1 ◽

1996 ◽

Vol 124 (6) ◽

pp. 1841-1846 ◽

Cited By ~ 1

Author(s):

William T. Ross

Keyword(s):

Dirichlet Space ◽

Invariant Subspaces ◽

Harmonic Dirichlet Space

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Multiplicities, invariant subspaces and an additive formula

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091521000146 ◽

2021 ◽

pp. 1-19

Author(s):

Arup Chattopadhyay ◽

Jaydeb Sarkar ◽

Srijan Sarkar

Keyword(s):

Hardy Space ◽

Invariant Subspace ◽

Dirichlet Space ◽

Invariant Subspaces ◽

Linear Operators ◽

Bounded Linear ◽

Generating Set ◽

Image Position ◽

Special Case ◽

Minimal Generating Set

Abstract Let $T = (T_1, \ldots , T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$ . The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$ . In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let $n \geq 2$ , and let $\mathcal{Q}_i$ , $i = 1, \ldots , n$ , be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb {C}$ . If $\mathcal{Q}_i^{\bot }$ , $i = 1, \ldots , n$ , is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\textbf {z}} = (M_{z_1}, \ldots , M_{z_n})$ -invariant subspace $(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{\perp }$ of the Dirichlet space or the Hardy space over the unit polydisc in $\mathbb {C}^{n}$ is given by \[ \mbox{mult}_{M_{\textbf{{z}}}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}}}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}} = \sum_{i=1}^{n} (\mbox{mult}_{M_z|_{\mathcal{Q}_i^{{\perp}}}} (\mathcal{Q}_i^{\bot})) = n. \] A similar result holds for the Bergman space over the unit polydisc.

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