Banach Spaces Of Analytic Functions Research Articles

Generalized Cesàro operators in weighted Banach spaces of analytic functions with sup-norms

AbstractAn investigation is made of the generalized Cesàro operators $$C_t$$ C t , for $$t\in [0,1]$$ t ∈ [ 0 , 1 ] , when they act on the space $$H({{\mathbb {D}}})$$ H ( D ) of holomorphic functions on the open unit disc $${{\mathbb {D}}}$$ D , on the Banach space $$H^\infty $$ H ∞ of bounded analytic functions and on the weighted Banach spaces $$H_v^\infty $$ H v ∞ and $$H_v^0$$ H v 0 with their sup-norms. Of particular interest are the continuity, compactness, spectrum and point spectrum of $$C_t$$ C t as well as their linear dynamics and mean ergodicity.

The Dynamic Behavior of Conjugate Multipliers on Some Reflexive Banach Spaces of Analytic Functions

In this paper, we characterize the hypercyclic, mixing and chaotic conjugate multipliers on some reflexive spaces of analytic functions. Our results extend several well-known results in the existing literature. We present some nontrivial examples to show the validity of our results.

Surjective Isometries of Multiplication Operators and Weighted Composition Operators on nth Weighted-Type Banach Spaces of Analytic Functions

This paper is to characterize isometries of multiplication operators and weighted composition operators on the nth weighted-type Banach spaces {Vn : n ∈ N} of analytic functions on the open unit disk of which the Bloch space and the Zygmund space are particular cases at n = 1, 2. We give characterizations of the symbols ψ and φ for which the multiplication operator Mψ and the weighted composition operator Wψ,φ are surjective isometries. Moreover, we show that generalized weighted composition operators are not isometric on Vn.

Pointwise multipliers between spaces of analytic functions

A Banach space X of analytic function in , the unit disc in , is said to be admissible if it contains the polynomials and convergence in X implies uniform convergence in compact subsets of . If X and Y are two admissible Banach spaces of analytic functions in and g is a holomorphic function in , g is said to be a multiplier from X to Y if g ·f ∈ Y for every f ∈ X. The space of all multipliers from X to Y is denoted M (X, Y), and M (X) will stand for M (X, X). The closed graph theorem shows that if g ∈ M (X, Y) then the multiplication operator M g, defined by Mg (f) = g · f, is a bounded operator from X into Y. It is known that M (X) ⊂ H ∞ and that if g ∈ M (X), then ∥g∥H ∞ ≤ ∥Mg∥. Clearly, this implies that M (X, Y) ⊂ H ∞ if Y ⊂ X. If Y ⊄ X, the inclusion M (X, Y) ⊂ H ∞ may not be true. In this paper we start presenting a number of conditions on the spaces X and Y which imply that the inclusion M (X, Y) ⊂ H ∞ holds. Next, we concentrate our attention on multipliers acting an BMOA and some related spaces, namely, the Qs-spaces (0<s<∞).

Essential norm of intrinsic operators from Banach spaces of analytic functions into weighted-type spaces

Essential norm of intrinsic operators from Banach spaces of analytic functions into weighted-type spaces

Product-type operators on a class of Banach spaces of analytic functions into $$\mu $$-Bloch spaces on the unit ball in $${\mathbb {C}}^n$$

Product-type operators on a class of Banach spaces of analytic functions into $$\mu $$-Bloch spaces on the unit ball in $${\mathbb {C}}^n$$

Surjective isometries on a Banach space of analytic functions with bounded derivatives

Let $$H(\mathbb D)$$ be the linear space of all analytic functions on the open unit disc $$\mathbb D$$ and $$H^p(\mathbb D)$$ the Hardy space on $$\mathbb D$$ . The characterization of complex linear isometries on $$\mathcal {S}^p=\{ f\in H(\mathbb D):f'\in H^p(\mathbb D) \}$$ was given for $$1 \le p < \infty $$ by Novinger and Oberlin in 1985. Here, we characterize surjective, not necessarily linear, isometries on $$\mathcal {S}^\infty $$ .

On the Cauchy dual operator and duality for Banach spaces of analytic functions

On the Cauchy dual operator and duality for Banach spaces of analytic functions

Generators of C0-semigroups of weighted composition operators

We prove that in a large class of Banach spaces of analytic functions in the unit disc ⅅ an (unbounded) operator Af = G · f′ + g · f with G, g analytic in ⅅ generates a C0-semigroup of weighted composition operators if and only if it generates a C0-semigroup. Particular instances of such spaces are the classical Hardy spaces. Our result generalizes previous results in this context and it is related to cocycles of flows of analytic functions on Banach spaces. Likewise, for a large class of non-separable Banach spaces X of analytic functions in ⅅ contained in the Bloch space, we prove that no non-trivial holomorphic flow induces a C0-semigroup of weighted composition operators on X. This generalizes previous results in [7] and [1] regarding C0-semigroup of (unweighted) composition operators.

Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

In this survey we report about recent work on weighted Banach spaces of analytic functions on the unit disc and on the whole complex plane defined with sup-norms and operators between them. Results about the solid hull and core of these spaces and distance formulas are reviewed. Differentiation and integration operators, Cesàro and Volterra operators, weighted composition and superposition operators and Toeplitz operators on these spaces are analyzed. Boundedness, compactness, the spectrum, hypercyclicity and (uniform) mean ergodicity of these operators are considered.

Invertible and Isometric Weighted Composition Operators

We consider abstract Banach spaces of analytic functions on general bounded domains that satisfy only a minimum number of axioms. We describe all invertible (equivalently, surjective) weighted composition operators acting on such spaces. For the spaces of analytic functions in the disk whose norm is given in terms of a natural seminorm (such as the typical spaces given in terms of derivatives), we describe all surjective weighted composition operators that are isometric. This generalizes a number of known results.

Approximation numbers of composition operators on weighted besov spaces of analytic functions

AbstractLi et al. [A spectral radius type formula for approximation numbers of composition operators, J. Funct. Anal. 267(12) (2014), 4753-4774] proved a spectral radius type formula for the approximation numbers of composition operators on analytic Hilbert spaces with radial weights and on $H^{p}$ spaces, $p\geq 1$, involving Green capacity. We prove that their formula holds for a wide class of Banach spaces of analytic functions and weights.

Sum of generalized weighted composition operators between weighted spaces of analytic functions

Let [Formula: see text] be the space of analytic functions on the unit disc [Formula: see text] and let [Formula: see text] denote the set of analytic self-maps of [Formula: see text]. Let [Formula: see text] be such that [Formula: see text] and [Formula: see text]. We characterize the boundedness, compactness and completely continuous of the sum of generalized weighted composition operators [Formula: see text] between weighted Banach spaces of analytic functions [Formula: see text] and [Formula: see text] which unifies the study of products of composition operators, multiplication operators and differentiation operators. As applications, we obtain the boundedness and compactness of the generalized weighted composition operators [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are weighted Bloch-type (little Bloch-type) spaces. Also, new characterizations of the boundedness and compactness of the operators [Formula: see text] and [Formula: see text] are given. Examples of bounded, unbounded, compact and non-compact operators [Formula: see text] and [Formula: see text] are given to explain the role of inducing functions [Formula: see text], [Formula: see text] and the weights [Formula: see text], [Formula: see text] of the underlying weighted spaces.

Product-type operators on Banach spaces of analytic functions on the upper half-plane

Let , the upper half-plane in the complex plane . Let be the set of all analytic functions on Π and be the set of all analytic self-maps on Π. For , and , where , , define an operator In this paper, we characterize the boundedness and compactness of the operator on the Hardy, weighted-Bergman spaces on Π to weighted-type spaces on Π.

Cyclic behaviour of certain operators on the disc algebra

We present results concerning the cyclic behaviour of weighted shifts and composition operators on the disc algebra. Most of our proofs use the representations of the dual space of the disc algebra as Banach spaces of analytic functions on the unit disc. Taking into account that no composition operator on the disc algebra can be hypercyclic, the results we obtain show that the composition operators induced by linear fractional maps with an interior fixed points have the same cyclic behaviour on the disc algebra as they do on the Hardy space , as described in Bourdon and Shapiro [Cyclic phenomena for composition operators. Mem Am Math Soc. 1997;125(596)], while the operators induced by linear fractional maps with boundary fixed points have the same cyclic behaviour on the disc algebra as they do on the Dirichlet space, as described in Gallardo-Gutiérrez and Montes-Rodríguez [The role of the spectrum in the cyclic behavior of composition operators. Mem Am Math Soc. 2004;167(791)].

Stability and collapse of the Lyapunov spectrum for Perron–Frobenius operator cocycles

In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron–Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron–Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That is, under perturbations, the Lyapunov exponents become 0 with multiplicity 1, and $-\\infty$ with infinite multiplicity. This is superficially similar to the finite-dimensional phenomenon, discovered by Bochi \[4], that away from the uniformly hyperbolic setting, small perturbations can lead to a collapse of the Lyapunov spectrum to zero. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different. We study stability of the Perron–Frobenius cocycles arising from general random Blaschke products. We give a necessary and sufficient criterion for stability of the Lyapunov spectrum in terms of the derivative of the random Blaschke product at its random fixed point, and use this to show that an open dense set of Blaschke product cocycles have hyperbolic Perron–Frobenius cocycles. In the final part, we prove a relationship between the Lyapunov spectrum of a single cocycle acting on two different Banach spaces, allowing us to draw conclusions for the same cocycles acting on $C^r$ function spaces.

Weighted conformal invariance of Banach spaces of analytic functions

We consider Banach spaces of analytic functions in the unit disc which satisfy a weighted conformal invariance property, that is, for a fixed α>0 and every conformal automorphism φ of the disc, f→f∘φ(φ′)α defines a bounded linear operator on the space in question, and the family of all such operators is uniformly bounded in operator norm. Many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces satisfy this condition. The aim of the paper is to develop a general approach to the study of such spaces based on this property alone. We consider polynomial approximation, duality and complex interpolation, we identify the largest and the smallest as well as the “unique” Hilbert space satisfying this property for a given α>0. We investigate the weighted conformal invariance of the space of derivatives, or anti-derivatives with the induced norm, and arrive at the surprising conclusion that they depend entirely on the properties of the (modified) Cesàro operator acting on the original space. Finally, we prove that this last result implies a John-Nirenberg type estimate for analytic functions g with the property that the integration operator f→∫0zf(t)g′(t)dt is bounded on a Banach space satisfying the weighted conformal invariance property.

Nuclear weighted composition operators on weighted Banach spaces of analytic functions

We characterize nuclear weighted composition operators W ψ , φ f = ψ ( f ∘ φ ) W_{\psi ,\varphi }f=\psi (f\circ \varphi ) on weighted Banach spaces of analytic functions with sup-norms. Consequences about nuclear composition operators on Bloch type spaces are presented. They extend previous work by Fares and Lefèvre on nuclear composition operators on Bloch spaces. Examples of nuclear as well as compact and non-nuclear weighted composition operators are given.

The problem of determining the 2D-kernel in a system of integro-differential equations of a viscoelastic porous medium

Under consideration is the system of integro-differential equations of a viscoelastic porous medium. The direct problem is to define the $$y$$ -component of the displacement vectors of the elastic porous body and the liquid from the initial boundary value problem for these equations. We assume that the kernel of the integral term of the first equation depends on time and one of the spatial variables. To determine the kernel, some additional condition is given on the solution of the direct problem for $$z=0 $$ . The inverse problem is replaced by an equivalent system of integro-differential equations for the unknown functions. We apply the method of scales of the Banach spaces of analytic functions. The local solvability of the inverse problem is proved in the class of the functions analytic in $$x$$ and continuous in $$t $$ .

Weighted composition operators in functional Banach spaces: an axiomatic approach

We work with very general Banach spaces of analytic functions in the disk or other domains which satisfy a minimum number of natural axioms. Among the preliminary results, we discuss some implications of the basic axioms and identify all functional Banach spaces in which every bounded analytic function is a pointwise multiplier. Next, we characterize (in various ways) the weighted composition operators among the bounded operators on such spaces, thus generalizing some well-known results on multiplication or composition operators. We also characterize the invertible weighted composition operators on the disk and on general Banach spaces of analytic functions on bounded domains under different sets of axioms whose connections we discuss by providing appropriate examples. This generalizes and complements various recent results by Gunatillake, Bourdon, and Hyv\"arinen-Lindstr\"om-Nieminen-Saukko.