Related papers: A $Z^k$-invariant subspace without the wandering p…

We show that in a scale of weighted Dirichlet spaces $D_{\alpha}$, including the Bergman space, given any finite Blaschke product $B$ there exists an equivalent norm in $D_{\alpha}$ such that $B$ satisfies the wandering subspace property…

Doubly commutativity of invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc $\mathbb{D}^n$ (with $ n \geq 2$) is investigated. We show that for any non-empty subset $\alpha=\{\alpha_1,\dots,\alpha_k\}$ of…

In this article, we define Dirichlet-type space $\mathcal{D}^{2}(\boldsymbol{\mu})$ over the bidisc $\mathbb D^2$ for any measure $\boldsymbol{\mu}\in\mathcal{P}\mathcal{M}_{+}(\mathbb T^2).$ We show that the set of polynomials is dense in…

The objective of this paper is to study wandering subspaces for commuting tuples of bounded operators on Hilbert spaces. It is shown that, for a large class of analytic functional Hilbert spaces $\mathcal{H}_K$ on the unit ball in $\mathbb…

We present two alternative proofs of Mandrekar's theorem, which states that an invariant subspaces of the Hardy space on the bidisc is of Beurling type precisely when the shifts satisfy a doubly commuting condition. The first proof uses…

We first generalize the results of Le\'on and M\"uller [Studia Math. 175(1) 2006] on hypercyclic subspaces to sequences of operators on Fr\'echet spaces with a continuous norm. Then we study the particular case of iterates of an operator T…

In this paper, subsystems with shadowing property for $\mathbb{Z}^{k}$-actions are investigated. Let $\alpha$ be a continuous $\mathbb{Z}^{k}$-action on a compact metric space $X$. We introduce the notions of pseudo orbit and shadowing…

It is proved that a commutative algebra $A$ of operators in a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when }…

In this paper, we use a new method to solve a long-standing problem. More specifically, we show that the Beurling-type theorem holds in the Bergman space $A^2_\alpha(D)$ for any $-1<\alpha < +\infty$. That is, every invariant subspace $H$…

We introduce and study Dirichlet-type spaces $\mathcal D(\mu_1, \mu_2)$ of the unit bidisc $\mathbb D^2,$ where $\mu_1, \mu_2$ are finite positive Borel measures on the unit circle. We show that the coordinate functions $z_1$ and $z_2$ are…

We consider random walks in Dirichlet environment (RWDE) on $\Z ^d$, for $ d \geq 3 $, in the sub-ballistic case. We associate to any parameter $ (\alpha_1, ..., \alpha_{2d}) $ of the Dirichlet law a time-change to accelerate the walk. We…

Given a commuting d-tuple $\bar T=(T_1,...,T_d)$ of otherwise arbitrary nonnormal operators on a Hilbert space, there is an associated Dirac operator $D_{\bar T}$. Significant attributes of the d-tuple are best expressed in terms of…

In this article we examine Dirichlet type spaces in the unit polydisc, and multipliers between these spaces. These results extend the corresponding work of G. D. Taylor in the unit disc. In addition, we consider functions on the polydisc…

Consider the Dirichlet-type space on the bidisk consisting of holomorphic functions $f(z_1,z_2):=\sum_{k,l\geq 0}a_{kl}z_1^kz_2^l$ such that $\sum_{k,l\geq 0}(k+1)^{\alpha_1} (l+1)^{\alpha_2}|a_{kl}|^2 <\infty.$ Here the parameters…

In this paper, we shall consider the killing transform induced by a multiplicative functional on regular Dirichlet subspaces of a fixed Dirichlet form. Roughly speaking, a regular Dirichlet subspace is a closed subspace with Dirichlet and…

Bishop operators $T_{\alpha}$ acting on $L^2[0,1)$ were proposed by E. Bishop in the fifties as possible operators which might entail counterexamples for the Invariant Subspace Problem. We prove that all the Bishop operators are…

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $\phi$ on the Dirichlet space $D$. We prove that any two distinct nontrivial minimal reducing subspaces of $M_\phi$ are orthogonal.…

For $\alpha \in \mathbb{R},$ we consider the scale of function spaces, namely the Dirichlet-type space $\mathcal{D}_{\alpha}$ consisting of holomorphic functions on the unit bidisk $\mathbb{D}^2$, $f(z,w)=\sum_{k,l=0}^{\infty}a_{kl}z^kw^l$…

We discuss a notion, originally introduced by Aleman in one variable, of Dirichlet-type space $\mathcal D(\mu_1,\mu_2)$ on the unit bidisc $\mathbb D^2,$ with superharmonic weights related to finite positive Borel measures $\mu_1,\mu_2$ on…

We study Dirichlet-type spaces $\mathfrak{D}_{\alpha}$ of analytic functions in the unit bidisk and their cyclic elements. These are the functions $f$ for which there exists a sequence $(p_n)_{n=1}^{\infty}$ of polynomials in two variables…