Consider the multiplication operator MB in $L^2(\mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products.

In this paper, we introduce the spherical polar decomposition of the linear pencil of an ordered pair $\mathbf {T}=(T_{1},T_{2})$ and investigate nontrivial invariant subspaces between the generalized spherical Aluthge transform of the linear pencil of $\mathbf {T}$ and the linear pencil of the original pair $\mathbf {T}$ of bounded operators with dense ranges.

Motivated by the near invariance of model spaces for the backward shift, we introduce a general notion of $(X,Y)$-invariant operators. The relations between this class of operators and the near invariance properties of their kernels are studied. Those lead to orthogonal decompositions for the kernels, which generalize well-known orthogonal decompositions of model spaces. Necessary and sufficient conditions for those kernels to be nearly X-invariant are established. This general approach can be applied to a wide class of operators defined as compressions of multiplication operators, in particular to Toeplitz operators and truncated Toeplitz operators, to study the invariance properties of their kernels (general Toeplitz kernels).

We characterize model polynomials that are cyclic in Dirichlet-type spaces in the unit ball of $\mathbb C^n$, and we give a sufficient capacity condition in order to identify noncyclic vectors.

We study the existence of reducing subspaces for rank-one perturbations of diagonal operators and, in general, of normal operators of uniform multiplicity one. As we will show, the spectral picture will play a significant role in order to prove the existence of reducing subspaces for rank-one perturbations of diagonal operators whenever they are not normal. In this regard, the most extreme case is provided when the spectrum of the rank-one perturbation of a diagonal operator $T=D + u\otimes v$ (uniquely determined by such expression) is contained in a line, since in such a case $T$ has a reducing subspace if and only if $T$ is normal. Nevertheless, we will show that it is possible to exhibit non-normal operators $T=D + u\otimes v$ with spectrum contained in a circle either having or lacking non-trivial reducing subspaces. Moreover, as far as the spectrum of $T$ is contained in any compact subset of the complex plane, we provide a characterization of the reducing subspaces $M$ of $T$ such that the restriction $T\mid _M$ is normal. In particular, such characterization allows us to exhibit rank-one perturbations of completely normal diagonal operators (in the sense of Wermer) lacking reducing subspaces. Furthermore, it determines completely the decomposition of the underlying Hilbert space in an orthogonal sum of reducing subspaces in the context of a classical theorem due to Behncke on essentially normal operators.

We consider a robust class of random non-uniformly expanding local homeomorphisms and Hölder continuous potentials with small variation. For each element of this class we develop the thermodynamical formalism and prove the existence and uniqueness of equilibrium states among non-uniformly expanding measures. Moreover, we show that these equilibrium states and the random topological pressure vary continuously in this setting.

Given a holomorphic self-map $\varphi $ of $\mathbb {D}$ (the open unit disc in $\mathbb {C}$ ), the composition operator $C_{\varphi } f = f \circ \varphi $ , $f \in H^2(\mathbb {\mathbb {D}})$ , defines a bounded linear operator on the Hardy space $H^2(\mathbb {\mathbb {D}})$ . The model spaces are the backward shift-invariant closed subspaces of $H^2(\mathbb {\mathbb {D}})$ , which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.

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

\[ \mbox{mult}_{M_{\textbf{{z}}}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}}}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}} = \sum_{i=1}^{n} (\mbox{mult}_{M_z|_{\mathcal{Q}_i^{{\perp}}}} (\mathcal{Q}_i^{\bot})) = n. \]

For an inner function u, we discuss the dual operator for the compressed shift $P_u S|_{{\mathcal {K}}_u}$, where ${\mathcal {K}}_u$ is the model space for u. We describe the unitary equivalence/similarity classes for these duals as well as their invariant subspaces.

We study ${{\text{w}}^{*}}$-semicrossed products over actions of the free semigroup and the free abelian semigroup on (possibly non-selfadjoint) ${{\text{w}}^{*}}$-closed algebras. We show that they are reflexive when the dynamics are implemented by uniformly bounded families of invertible row operators. Combining with results of Helmer, we derive that ${{\text{w}}^{*}}$-semicrossed products of factors (on a separableHilbert space) are reflexive. Furthermore, we show that ${{\text{w}}^{*}}$-semicrossed products of automorphic actions on maximal abelian self adjoint algebras are reflexive. In all cases we prove that the ${{\text{w}}^{*}}$-semicrossed products have the bicommutant property if and only if the ambient algebra of the dynamics does also.

In this paper, we develop a generalized Jordan canonical form theorem for a certain class of operators in $L\left( H \right)$ . A complete criterion for similarity for this class of operators in terms of $K$ -theory for Banach algebras is given.

In this paper we generalise a result of Izuchi and Suárez (K. Izuchi and D. Suárez, Norm-closed invariant subspaces in L∞ and H∞, Glasgow Math. J. 46 (2004), 399–404) on the shift invariant subspaces of $L^\infty(\mathbb{T})$ to the non-commutative setting. Considering these subspaces as $C(\mathbb{T})$-modules contained in $L^\infty(\mathbb{T})$, we show that under some restrictions, a similar description can be given for the ${\mathfrak{B}}$-submodules of ${\mathfrak{A}}$, where ${\mathfrak{A}}$ is a C*-algebra and ${\mathfrak{B}}$ is a commutative C*-subalgebra of ${\mathfrak{A}}$. We use this to give a description of the $\mathbb{M}_n({\mathfrak{B}})$-submodules of $\mathbb{M}_n({\mathfrak{A}})$.

We obtain a complete description of closed ideals of the algebra $\mathcal{D}\cap \text{li}{{\text{p}}_{\alpha }},0<\alpha \le \frac{1}{2}$, where $\mathcal{D}$ is the Dirichlet space and $\text{li}{{\text{p}}_{\alpha }}$ is the algebra of analytic functions satisfying the Lipschitz condition of order $\alpha $.

It is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A. C. M. van Rooij and W. H. Schikhof in 1992.

An operator A on a complex, separable, infinite-dimensional Hilbert space H is hypercyclic if there is a vector x∈H such that the orbit {x,Ax,A2x,…} is dense in H. Using the character of the analytic core and quasinilpotent part of an operator A, we explore the hypercyclicity for upper triangular operator matrix

This paper is a continuation of three recent articles concerning the structure of hyperinvariant subspace lattices of operators on a (separable, infinite dimensional) Hilbert space $\mathcal{H}$. We show herein, in particular, that there exists a “universal” fixed block-diagonal operator $B$ on $\mathcal{H}$ such that if $\varepsilon >0$ is given and $T$ is an arbitrary nonalgebraic operator on $\mathcal{H}$, then there exists a compact operator $K$ of norm less than $\varepsilon $ such that (i) Hlat$(T)$ is isomorphic as a complete lattice to Hlat$(B+K)$ and (ii) $B+K$ is a quasidiagonal, ${{C}_{00}}$, $(\text{BCP})$-operator with spectrum and left essential spectrum the unit disc. In the last four sections of the paper, we investigate the possible structures of the hyperlattice of an arbitrary algebraic operator. Contrary to existing conjectures, Hlat$(T)$ need not be generated by the ranges and kernels of the powers of $T$ in the nilpotent case. In fact, this lattice can be infinite.

We give bounds on the distance from a non-zero idempotent to the set of nilpotents in the set of $n\,\times \,n$ matrices in terms of the norm of the idempotent. We construct explicit idempotents and nilpotents which achieve these distances, and determine exact distances in some special cases.

The Banach convolution algebras ${{l}^{1}}(\omega )$ and their continuous counterparts ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$ are much studied, because (when the submultiplicative weight function $\omega $ is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of “nice” weights $\omega $, the only closed ideals they have are the obvious, or “standard”, ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in ${{l}^{1}}(\omega )$. His proof was successfully exported to the continuous case ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$ by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in ${{l}^{1}}(\omega )$ and ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$. The new proof is based on the idea of a “nonstandard dual pair” which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$ containing functions whose supports extend all the way down to zero in ${{\mathbb{R}}^{+}}$, thereby solving what has become a notorious problem in the area.

In [11] the authors obtained an operator matrix with two variables that distinguishes the classes of $p$-hyponormal operators, $w$-hyponormal, absolute-$p$-paranormal, and normaloid operators on Hilbert spaces. We establish the general model for $n$ variables, which provides many more examples to show that such classes are distinct.

Let $L^{2}=L^{2}(D,rdrd\theta/\pi)$ be the Lebesgue space on the open unit disc $D$ and let $L_{a}^2=L^{2}\cap\mathrm{Hol}(D)$ be a Bergman space on $D$. In this paper, we are interested in a closed subspace $\mathcal{M}$ of $L^{2}$ which is invariant under the multiplication by the coordinate function $z$, and a Hankel-type operator from $L_{a}^2$ to $\mathcal{M}^\bot$. In particular, we study an invariant subspace $\mathcal{M}$ such that there does not exist a finite-rank Hankel-type operator except a zero operator.