Sobolev spaces in several variables inL 1-type norms are not isomorphic to Banach lattices

A Sobolev space in several variables in anL ¹-type norm is not complemented in its second dual. Hence it is not isomorphic as a Banach space to any complemented subspace of a Banach lattice. Both authors were supported in part by the Polish KBN grant 2P03A 036 14.     

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.     

Invariant subspaces of finite codimension in Banach spaces of analytic functions

We give a characterization of invariant subspaces of finite codimension in Banach spaces of vector-valued analytic functions in several variables, where invariant refers to invariance under multiplication by any polynomial. We obtain very weak conditions under which our characterization applies, that unifies and improves a number of previous results. In the vector-valued case, the results are new even for one complex variable. As a concrete application in several variables, we consider the Bergman space on a strictly pseudo-convex domain, and we improve previous results (assuming C^∞-boundary) to the case of C²-boundary.     

Boundaries for algebras of analytic functions on function module Banach spaces

We consider the uniform algebra of continuous and bounded functions that are analytic on the interior of the closed unit ball of a complex Banach function module X. We focus on norming subsets of , i.e., boundaries, for such algebra. In particular, if X is a dual complex Banach space whose centralizer is infinite‐dimensional, then the intersection of all closed boundaries is empty. This also holds in case that X is an ‐sum of infinitely many Banach spaces and further, the torus is a boundary.     

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.     

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.     

Superposition operators between weighted Banach spaces of analytic functions of controlled growth

We characterize the entire functions which transform a weighted Banach space of holomorphic functions on the disc of type $H^{\infty }$ into another such space by superposition. We also show that all the superposition operators induced by such entire functions map bounded sets into bounded sets and are continuous. Superposition operators that map bounded sets into relatively compact sets are also considered.     

Retracting a ball onto a sphere in some Banach spaces

The unit sphere in an infinite dimensional Banach space is a lipschitzian retract of the unit ball. The aim of this paper is to present a new upper bound for the optimal retraction constant in some classical Banach spaces. In particular, an improved estimate from above is obtained for the space C[0,1].     

A method of constructing conditional bases of power functions in certain Banach spaces of analytic functions

In this note, a, method of constructing conditional exponential bases in some Banach spaces of analytic functions is presented. The method of constructing basic families due to Figà-Talamanca is generalized. Concrete applications and examples of conditional basic families in the space of multipliers of power series with the sequence of Taylor coefficients from l^p, in the space of multipliers of Cauchy-type integrals, and in other spaces are given. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 170–199. Translated by S. V. Kislyakov.     

SeparableG spaces are isomorphic toC(K) spaces

We show that a wide class of separable preduals ofL ¹(μ) spaces, namely, theG spaces, introduced by Grothendieck, are isomorphic toC(K) spaces. This is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss. I wish to thank Professor Lindenstrauss for his interest and advice.     

Spectra and essential spectral radii of composition operators on weighted Banach spaces of analytic functions

We determine the spectra of composition operators acting on weighted Banach spaces of analytic functions on the unit disc defined for a radial weight v, when the symbol of the operator has a fixed point in the open unit disc. We also investigate in this case the growth rate of the Koenigs eigenfunction and its relation with the essential spectral radius of the composition operator.     

Bump functions and differentiability in Banach spaces

We show that if a Banach space admits a continuous symmetrically Fréchet subdifferentiable bump function, then is an Asplund space.