Isometries between Banach spaces of Lipschitz functions

The Banach spaces Lip ^a (S, Δ), lip ^a (S, Δ), Lip ^a (S, Δ;s ₀) and lip ^a (S, Δ;s ₀) of Lipschitz functions are defined. We shall identify the extreme points of the unit balls in their corresponding dual spaces and make use of them to present a complete characterization of the isometries between these function spaces. This paper is a part of the author’s M.Sc. thesis which was prepared under the guidance of Dr. Y. Benyamini.     

Multiplication operators on Hilbert spaces of analytic functions

Let be a Hilbert space of functions analytic on a plane domain such that for every in the functional of evaluation at is bounded. Assume further that contains the constants and admits multiplication by the independent variable z, M_z, as a bounded operator. We give sufficient conditions for M_z to be reflexive.Received: 17 February 2004     

On the optimal approximation of bounded linear functionals in Hilbert spaces of analytic functions

Optimal numerical approximation of bounded linear functionals by weighted sums in Hilbert spaces of functions analytic in a circleK _r , in a circular annulusK _r1,r2 and in an ellipseE _r is investigated by Davis' method on the common algebraic background for diagonalising the normal equation matrix. The weights and error functional norms for optimal rules with nodes located angle-equidistant on the concentric circleK _s or on the confocal ellipseE _s and in the interval [–1,1] for an arbitrary bounded linear functional are given explicitly. They are expressed in terms of a complete orthonormal system in the Hilbert space.     

Interpolation in certain Hilbert spaces of analytic functions

In this article we study, for a Hilbert spaceB of analytic functions in the open unit disk, the dependence of the structure of the space of sequencesB(Z)={{f(z_k)} _k=1 ^∞ :f∈B} on the choice of the sequence Z={z_k} _k=1 ^∞ of distinct points of the unit disk [6].     

Isometries of some Banach spaces of analytic functions

We characterize the surjective isometries of a class of analytic functions on the disk which include the Analytic Besov spaceB _p and the Dirichlet spaceD ^p . In the case ofB _p we are able to determine the form of all linear isometries on this space. The isometries for these spaces are finite rank perturbations of integral operators. This is in contrast with the classical results for the Hardy and Bergman spaces where the isometries are represented as weighted compositions induced by inner functions or automorphisms of the disk.     

Perturbed smooth Lipschitz extensions of uniformly continuous functions on Banach spaces

We show that if is a separable subspace of a Banach space such that both and the quotient have -smooth Lipschitz bump functions, and is a bounded open subset of , then, for every uniformly continuous function and every 0$">, there exists a -smooth Lipschitz function such that for every .

     

Adjoints of composition operators on Hilbert spaces of analytic functions

We observe that a formula for the adjoint of a composition operator, known only for special symbols in some spaces of analytic functions, actually holds for every admissible symbol and in any Hilbert space of analytic functions with reproducing kernels. Along with some new results, all known formulas for the adjoint obtained so far follow easily as a consequence, some in an improved form.     

Compact composition operators on weighted Hilbert spaces of analytic functions

We characterize the compactness of composition operators acting on a large family of Hilbert spaces of analytic functions which lie between Bergman and Dirichlet spaces. Our characterization is given in terms of generalized Nevanlinna counting functions.     

Cluster values of analytic functions on a Banach space

We investigate uniform algebras of bounded analytic functions on the unit ball of a complex Banach space. We prove several cluster value theorems, relating cluster sets of a function to its range on the fibers of the spectrum of the algebra. These lead to weak versions of the corona theorem for ? ₂ and for c ₀. In the case of the open unit ball of c ₀, we solve the corona problem whenever all but one of the functions comprising the corona data are uniformly approximable by polynomials in functions in ${c_0^*}$ .     

Composition operators on a Banach space of analytic functions

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL _α ¹ (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range. Supported by NNSFC No.19671036     

Smooth approximation of convex functions in Banach spaces

This paper shows that every extended-real-valued lower semi-continuous proper (respectively Lipschitzian) convex function defined on an Asplund space can be represented as the point-wise limit (respectively uniform limit on every bounded set) of a sequence of Lipschitzian convex functions which are locally affine (hence, C^∞) at all points of a dense open subset; and shows an analogous for w^∗-lower semi-continuous proper (respectively Lipschitzian) convex functions defined on dual spaces whose pre-duals have the Radon-Nikodym property.     

Boundary behavior in Hilbert spaces of vector-valued analytic functions

In this paper we study the boundary behavior of functions in Hilbert spaces of vector-valued analytic functions on the unit disc D. More specifically, we give operator-theoretic conditions on Mz, where Mz denotes the operator of multiplication by the identity function on D, that imply that all functions in the space have non-tangential limits a.e., at least on some subset of the boundary. The main part of the article concerns the extension of a theorem by Aleman, Richter and Sundberg in [A. Aleman, S. Richter, C. Sundberg, Analytic contractions and non-tangential limits, Trans. Amer. Math. Soc. 359 (2007)] to the case of vector-valued functions.     

Fredholm composition operators on algebras of analytic functions on Banach spaces

We prove that Fredholm composition operators acting on the uniform algebra H^∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.     

Removable singularities for analytic functions in BMO and locally Lipschitz spaces

In this paper we study removable singularities for holomorphic functions such that sup_z|f⁽ⁿ⁾(z)|dist(z,)^s<. Spaces of this type include spaces of holomorphic functions in Campanato classes, BMO and locally Lipschitz classes. Dolzhenko (1963), Král (1976) and Nguyen (1979) characterized removable singularities for some of these spaces. However, they used a different removability concept than in this paper. They assumed the functions to belong to the function space on and be holomorphic on \ E, whereas we only assume that the functions belong to the function space on \ E, and are holomorphic there. Koskela (1993) obtained some results for our type of removability, in particular he showed the usefulness of the Minkowski dimension. Kaufman (1982) obtained some results for s=0.In this paper we obtain a number of examples with certain important properties. Similar examples have earlier been obtained for Hardy H^p classes and weighted Bergman spaces, mainly by the author. Because of the similarities in these three cases, an axiomatic approach is used to obtain some results that hold in all three cases with the same proofs. Supported by the Swedish Research Council and Gustaf Sigurd Magnusons fund of the Royal Swedish Academy of Sciences.Mathematics Subject Classification (2000):30B40, 30D45, 30D55, 46E15.     

A simple proof of the approximation by real analytic Lipschitz functions

A theorem in Azagra et al. (preprint) [1] asserts that on a real separable Banach space with separating polynomial every Lipschitz function can be uniformly approximated by real analytic Lipschitz function with a control over the Lipschitz constant. We give a simple proof of this theorem.     

Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions

We investigate weighted composition operators that attain their norm on weighted Banach spaces of holomorphic functions on the unit disc of type H ^∞. Applications for composition operators on weighted Bloch spaces are given.