If $b$ is an inner function, then composition with $b$ induces an endomorphism, $β$, of $L^\infty(\mathbb{T})$ that leaves $H^\infty(\mathbb{T})$ invariant. We investigate the structure of the endomorphisms of $B(L^2(\mathbb{T}))$ and $B(H^2(\mathbb{T}))$ that implement $β$ through the representations of $L^\infty(\mathbb{T})$ and $H^\infty(\mathbb{T})$ in terms of multiplication operators on $L^2(\mathbb{T})$ and $H^2(\mathbb{T})$. Our analysis, which is based on work of R. Rochberg and J. McDonald, will wind its way through the theory of composition operators on spaces of analytic functions to recent work on Cuntz families of isometries and Hilbert $C^*$-modules.
We investigate properties of essential spectra of disjointness preserving operators acting on Banach $C(K)$-modules. In particular, we prove that under some very mild conditions the upper semi-Fredholm spectrum of such an operator is rotation invariant. In the last part of the paper we provide a full description of the spectrum and the essential spectra of operators acting on Kaplansky modules of the form $T = wU$, where $w \in C(K)$, $U$ is a $d$-isomorphism, and the spectrum of $U$ is a subset of the unit circle.
We develop a new duality between endomorphisms of measure spaces, on the one hand, and a certain family of positive operators, called transfer operators, acting in spaces of measurable functions on, on the other. A framework of standard Borel spaces is adopted; and this generality is wide enough to cover a host of applications. While the mathematical structures of positive operators, endomorphisms, transfer operators, measurable partitions, and Markov processes arise in a host of settings, both pure and applied, we propose here a unified study. This is the general setting of dynamics in Borel …
The aim of this work is to study the structure of bounded finite potent endomorphisms on Hilbert spaces. In particular, for these operators, an answer to the Invariant Subspace Problem is given and the main properties of its adjoint operator are offered. Moreover, for every bounded finite potent endomorphism we show that Tate's trace coincides with the Leray trace and with the trace defined by R. Elliott for Riesz Trace Class operators.
The aim of this paper is to characterize continuous endomorphisms in the space of entire functions of exponential type of order $p>0$. Let $A_p$ denote the space of entire functions of $n$ complex variables $z\in{\mathbb C}^n$ of order $p$ of normal type. We consider an endomorphism $F$ in the space, which is considered to be a DFS-space. We show that there is a unique linear differential operator $P$ of infinite order with coefficients in the space which realizes $F$, that is, $Ff=Pf$ holds for any $f\in A_p$. The coefficients satisfy certain growth conditions and conversely, if a formal diff…
We study the bounded endomorphisms of $\ell_{N}^2(G)=\ell^2(G)\times \dots \times\ell^2(G)$ that commute with translations, where $G$ is a discrete abelian group. It is shown that they form a C*-algebra isomorphic to the C*-algebra of $N\times N$ matrices with entries in $L^\infty(\widehat{G})$, where $\widehat{G}$ is the dual space of $G$. Characterizations of when these endomorphisms are invertible, and expressions for their norms and for the norms of their inverses, are given. These results allow us to study Riesz systems that arise from the action of $ G $ on a finite set of elements of a …
We study positive transfer operators $R$ in the setting of general measure spaces $\left(X,\mathscr{B}\right)$. For each $R$, we compute associated path-space probability spaces $\left(Ω,\mathbb{P}\right)$. When the transfer operator $R$ is compatible with an endomorphism in $\left(X,\mathscr{B}\right)$, we get associated multiresolutions for the Hilbert spaces $L^{2}\left(Ω,\mathbb{P}\right)$ where the path-space $Ω$ may then be taken to be a solenoid. Our multiresolutions include both orthogonality relations and self-similarity algorithms for standard wavelets and for generalized wavelet-res…
We examine the semicrossed products of a semigroup action by $*$-endomorphisms on a C*-algebra, or more generally of an action on an arbitrary operator algebra by completely contractive endomorphisms. The choice of allowable representations affects the corresponding universal algebra. We seek quite general conditions which will allow us to show that the C*-envelope of the semicrossed product is (a full corner of) a crossed product of an auxiliary C*-algebra by a group action. Our analysis concerns a case-by-case dilation theory on covariant pairs. In the process we determine the C*-envelope fo…
In this paper we deal with unbounded composition operators defined in Orlicz spaces. Indeed, we provide some necessary and sufficient condition for densely definedness of composition operators on Orlicz spaces. Also, we will investigate the adjoint of densely defined composition operators and we give some equivalent conditions for it to be densely defined. In addition, we show that densely defined composition operator is continuous if and only if it is everywhere defined. Finally, we characterize densely defined continuous composition operators.
Given a symbol $φ,$ i.e., a holomorphic endomorphism of the unit disc, we consider the composition operator $C_φ(f)=f\circφ$ defined on the Banach spaces of holomorphic functions $A(\mathbb{D})$ and $H^{\infty}(\mathbb{D})$. We obtain different conditions on the symbol $φ$ which characterize when the composition operator is mean ergodic and uniformly mean ergodic in the corresponding spaces. These conditions are related to the asymptotic behaviour of the iterates of the symbol. As an appendix, we deal with some particular case in the setting of weighted Banach spaces of holomorphic functions.
In this paper we study the self-adjoint and co-isometry composition and weighted composition operator on $H_E(ζ).$ We also discuss the conditions under which adjoint operator of a composition and weighted composition operator on $H_E(ζ)$ to be some composition and weighted composition operator, respectively.
In this paper we obtain characterizations for adjoint of a composition and weighted composition operator to be composition and weighted composition operator on $F_ψ^2,$ respectively. We study the co-isometry composition and weighted composition operators on $F_ψ^2.$
In this paper, we consider the locally convex spaces of entire functions with growth given by proximate orders, and study the representation as a differential operator of a continuous homomorphism from such a space to another one. As a corollary, we give a characterization of continuous endomorphisms of such spaces.
Let $S(\mathbb{D})$ be the collection of all holomorphic self-maps on $\mathbb{D}$ of the complex plane $\mathbb{C}$, and $C_φ$ the composition operator induced by $φ\in S(\mathbb{D})$. We obtain that there are no hypercyclic composition operators on the little Bloch space $\mathcal{B}_0$ and the Besov space $B_p$.
We provide a complete description of the spectrum and the essential spectra of weighted composition operators $T=wT_φ$ on $C(K)$ in the case when the map $φ$ is a non-invertible homeomorphism of $K$ into itself.
In this paper, we study weighted composition operators on the Fock space. We show that a weighted composition operator is cohyponorma if and only if it is normal. Moreover, we give a complete characterization of closed range weighted composition operators. Finally, we find norms of some weighted composition operators.
In this paper we consider composition operators on Harmonic-Bloch type spaces and we compute the spectrum of composition operators. Also, we characterize isometric composition operators on harmonic Bloch type spaces.
We study a family of discrete-time random-walk models. The starting point is a fixed generalized transfer operator $R$ subject to a set of axioms, and a given endomorphism in a compact Hausdorff space $X$. Our setup includes a host of models from applied dynamical systems, and it leads to general path-space probability realizations of the initial transfer operator. The analytic data in our construction is a pair $\left(h,λ\right)$, where $h$ is an $R$-harmonic function on $X$, and $λ$ is a given positive measure on $X$ subject to a certain invariance condition defined from $R$. With this we sh…
We obtain necessary and sufficient conditions for the composition and weighted composition operator and product of composition operators to be isometry and unitary on $H_{E}(ξ).$ With the help of counter example we also prove that the product of two non isometric composition operator and weighted composition operator can be isometry on $H_{E}(ξ)$. We also completely characterize the boundedness of generalized weighted composition operators on $H_{E}(ξ).$
In this paper we find all complex symmetric weighted composition operators with special conjugations. Then we give spectral properties of these complex symmetric weighted composition operators.
We study a composition operator on Lorentz spaces. In particular we provide necessary and sufficient conditions under which a measurable mapping induces a bounded composition operator.
Unbounded composition operators in $L^2$-space over discrete measure spaces are investigated. Normal, formally normal and quasinormal composition operators acting in $L^2$-spaces of this kind are characterized.
We study composition operators on the Schwartz space of rapidly decreasing functions. We prove that such a composition operator is never a compact operator and we obtain necessary or sufficient conditions for the range of the composition operator to be closed. These conditions are expressed in terms of multipliers for the Schwartz class and the closed range property of the corresponding operator considered in the space of smooth functions.
We study composition operators on spaces of double Dirichlet series, focusing our interest on the characterization of the composition operators of the space of bounded double Dirichlet series $\HCdos$. We also show how the composition operators of this space of Dirichlet series are related to the composition operators of the corresponding spaces of holomorphic functions. Finally, we give a characterization of the superposition operators in $\HC$ and in the spaces $\mathcal{H}^p$.
We study the weighted composition operators between the Lipschitz space and the space of bounded functions on the set of vertices of an infinite tree. We characterized the boundedness, the compactness, and the boundedness from below of weighted composition operators. We also determine the isometric weighted composition operators.
In this paper we characterize essential norm of composition operators on the spaces of Harmonic Bloch functions. These results extends the similar results that were proven for composition operators on Bloch spaces.
We prove the subnormality of an operator-weighted composition operator whose symbol is a transformation of a discrete measure space and weights are multiplication operators in $L^2$-spaces under the assumption of existence of a family of probability measures whose Radon-Nikodym derivatives behave regular along the trajectories of the symbol. We build the quasinormal extension which is a weighted composition operator induced by the same symbol. We give auxiliary results concerning commutativity of operator-weighted composition operators with multiplication operators.
We study composition operators on the weighted Banach spaces of an infinite tree. We characterize the bounded and the compact operators, as well as determine the operator norm and the essential norm. In addition, we study the isometric composition operators.
In this paper, we investigate weighted composition, Volterra and Integral operators on second derivative Hardy spaces. Some equivalent conditions for boundedness of the operators will be given using the boundedness on the Hardy spaces. Also we give a criteria for compactness of weighted composition operators.
We show that the symbol of a bounded composition operator on the Wiener algebra of Dirichlet series does not need to belong to this algebra. Our example even gives an absolutely summing (hence compact) composition operator.
Bounded weighted composition operators, as well as compact weighted composition operators, on Fock spaces have been characterised. This characterisation is refined to the extent that the question of whether weighted composition operators on the Fock space can be supercyclic is answered in the negative.
We consider the products of composition and differentiation operators on the Hardy space. We provide a complete characterization of the boundedness and compactness of these operators. Furthermore, we obtain the explicit condition for these operators to be Hilbert-Schmidt operators.
We characterize lower semi-Fredholm and Fredholm of weighted composition operators on $C(K)$ in the case when the corresponding map is an open surjection of the compact space $K$ onto itself. The obtained criterions involve the notion of almost homeomorphisms of compact spaces.
The paper deals with unbounded composition operators with infinite matrix symbols acting in $L^2$-spaces with respect to the gaussian measure on $\mathbb{R}^\infty$. We introduce weak cohyponormality classes $\EuScript{S}_{n,r}^*$ of unbounded operators and provide criteria for the aforementioned composition operators to belong to $\EuScript{S}_{n,r}^*$. Our approach is based on inductive limits of operators.
In this paper we characterize some basic properties of composition operators on the spaces of harmonic Bloch functions. First we provide some equivalent conditions for boundedness and compactness of composition operators. Then by using these conditions we estimate the essential norm of composition operators. These results extends the similar results that were proven in the literature on the Bloch spaces.
The properties of Volterra-composition operators on the weighted Bergman space with exponential type weights are investigated in this paper. We state some necessary and sufficient conditions that a Volterra-composition operator from the weighted Bergman space to Bloch type space ( or little Bloch type space ) must satisfy for the Volterra-composition operator to be bounded or compact.
In this paper, we study the weighted composition operator on the Fock space $\mf$ of slice regular functions. First, we characterize the boundedness and compactness of the weighted composition operator. Subsequently, we describe all the isometric composition operators. Finally, we introduce a kind of (right)-anti-complex-linear weighted composition operator on $\mf$ and obtain some concrete forms such that this (right)-anti-linear weighted composition operator is a (right)-conjugation. Specially, we present equivalent conditions ensuring weighted composition operators which are conjugate $\mat…
We show that the generalised composition of generalised integral operators is well defined on the space Colombeau algebras of tempered generalised functions.
We provide a description of the spectrum and essential spectra of invertible weighted composition operators acting in some algebras of smooth functions on the interval.
In this paper, we study quasinormal and hyponormal composition operators \W with linear fractional compositional symbol $\ph$ on the Hardy and weighted Bergman spaces. We characterize the quasinormal composition operators induced on $H^{2}$ and $A_α^{2}$ by these maps and many such weighted composition operators, showing that they are necessarily normal in all known cases. We eliminate several possibilities for hyponormal weighted composition operators but also give new examples of hyponormal weighted composition operators on $H^2$ which are not quasinormal.
The kernel of composition operator $C_T$ on Orlicz-Sobolev space is obtained. Using the kernel, a necessary and a sufficient condition for injectivity of composition operator $C_T$ has been established. Composition operators on Orlicz-Sobolev space with finite ascent as well as infinite ascent have been characterized.
This paper characterises the boundedness and compactness of Agler--McCarthy monomial operators by reducing them to weighted composition operators and deriving explicit Carleson measure criteria on the half-plane. The results are illustrated by examples.
