Invariant subspaces of the Dirichlet space and commutative algebra

2004 ◽  
Vol 2004 (569) ◽  
Author(s):  
Xiang Fang
Keyword(s):  
Commutative Algebra ◽  
Dirichlet Space ◽  
Invariant Subspaces

Bergman Spaces on Disconnected Domains

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.

scholarly journals Invariant subspaces of parabolic self-maps in the Dirichlet space

2014 ◽  
Vol 266 (7) ◽  
pp. 4115-4120
Author(s):  
Alfonso Montes-Rodríguez ◽  
Manuel Ponce-Escudero
Keyword(s):  
Dirichlet Space ◽  
Invariant Subspaces

Invariant subspaces of the Dirichlet space

2010 ◽  
pp. 133-141 ◽  
Author(s):  
Omar El-Fallah ◽  
Karim Kellay ◽  
Thomas Ransford
Keyword(s):  
Dirichlet Space ◽  
Invariant Subspaces

scholarly journals Multiplicities, invariant subspaces and an additive formula

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.

scholarly journals On unitary equivalence of invariant subspaces of the Dirichlet space

2010 ◽  
Vol 196 (2) ◽  
pp. 143-150 ◽  
Author(s):  
Kunyu Guo ◽  
Liankuo Zhao
Keyword(s):  
Dirichlet Space ◽  
Invariant Subspaces ◽  
Unitary Equivalence

Hankel operators and invariant subspaces of the Dirichlet space

2015 ◽  
Vol 91 (2) ◽  
pp. 423-438 ◽  
Author(s):  
Shuaibing Luo ◽  
Stefan Richter
Keyword(s):  
Dirichlet Space ◽  
Invariant Subspaces ◽  
Hankel Operators

Invariant Subspaces on KPC-Hypergroups

2019 ◽  
Vol 15 (1) ◽  
pp. 122-130
Author(s):  
Laszlo Szekelyhidi ◽  
◽  
Seyyed Mohammad Tabatabaie ◽  
Keyword(s):  
Invariant Subspaces
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