Weighted composition operators between weighted spaces of vector-valued holomorphic functions on Banach spaces

Let B(E, F) be the Banach Space of all continuous linear operators from a Banach Space E into a Banach space F. Let U_X and U_Y be balanced open subsets of Banach spaces X and Y, respectively. Let V and W be two Nachbin families of continuous weights on U_X and U_Y, respectively. Let HV(U_X, E) (or HV₀(U_X, E)) and HW(U_Y, F) (or HW₀(U_Y, F)) be the weighted spaces of vector-valued holomorphic functions. In this paper, we investigate the holomorphic mappings ? : U_Y → U_X and ψ : U_Y → B(E, F) which generate weighted composition operators between these weighted spaces.     

Weighted composition operators on Hilbert spaces of vector-valued functions

In this paper we investigate criterion for the hyponormality, cohyponormality, and normality of weighted composition operators acting on Hilbert spaces of vector-valued functions.

     

Composition operators acting between hardy spaces

A function-theoretic necessary and sufficient condition on a symbol is given for the compactness of the induced composition operator acting betweenH ^p andH ^q , forq. Compact differences of such composition operators are shown to occur only in the trivial case of both operators being compact themselves.     

Hardy spaces of vector-valued functions

The purpose of this paper is to consider the relations among the various definitions of the spaces H^p of analytic functions with values in a Banach space and to investigate the problem of the structure of the conjugates of these spaces. In particular, one constructs an example of a reflexive separable Banach space χ, for which the equality H^p(X)^*=H^p′(X^*) (1<p<∞,1/p+1/p′=1) ails. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 5–16, 1976.     

Sobolev spaces of vector-valued functions

Generalizations of Sobolev spaces in two directions are considered: firstly, instead of LP-norms one makes use of more general norms and, secondly, spaces of functions with values in a Banach space are investigated. The problem of the connection between such spaces and spaces of Bessel potentials is solved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 5–14, 1991.     

Adjoints of semigroups acting on vector-valued function spaces

LetT(t) be the translation group onY=C ₀(ℝ×K)=C ₀(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY ^⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y ^⊙=L ¹(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT ₀(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT ₀(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ ^⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C ₀(Ω;X) andL ^p (μ;X). This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful to the CWI and the Dutch National Science Foundation NWO for financial support.     

Weighted spaces of vector-valued continuous functions

Summary We study completeness of weighted spaces of vector-valued functions. Also, the dual of a weighted space of vector-valued continuous functions on a locally compact space is characterized as a space of vector valued Radon measures with values in the topological dual of the range space. Research supported by a grant from the National Science Foundation, GP 22713. Entrata in Redazione il 25 ottobre 1970.     

Local dual spaces of Banach spaces of vector-valued functions

We show that is a local dual of , and is a local dual of , where is a Banach space. A local dual space of a Banach space is a subspace of so that we have a local representation of in satisfying the properties of the representation of in provided by the principle of local reflexivity.

     

Order-weakly compact operators from vector-valued function spaces to Banach spaces

Let be an ideal of over a  -finite measure space , and let stand for the order dual of . For a real Banach space let be a subspace of the space of -equivalence classes of strongly -measurable functions and consisting of all those for which the scalar function belongs to . For a real Banach space a linear operator is said to be order-weakly compact whenever for each the set is relatively weakly compact in . In this paper we examine order-weakly compact operators . We give a characterization of an order-weakly compact operator in terms of the continuity of the conjugate operator of with respect to some weak topologies. It is shown that if is an order continuous Banach function space, is a Banach space containing no isomorphic copy of and is a weakly sequentially complete Banach space, then every continuous linear operator is order-weakly compact. Moreover, it is proved that if is a Banach function space, then for every Banach space any continuous linear operator is order-weakly compact iff the norm is order continuous and is reflexive. In particular, for every Banach space any continuous linear operator is order-weakly compact iff is reflexive.

     

Boundedness of vector-valued operators on weighted Herz spaces

In this paper, the authors establish the weighted weak and strong type norm inequalities in the set of weighted Herz-type spaces for a vector-valued analogue of the Hardy-Littlewood maximal operator; and using this, the authors obtain the weightd inequalies for a wide class of sublinear singular operators defined onR ⁿ which include the Calderón-Zygmund operators as special cases. The fractional versions of these results are also given.     

Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions

We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if Ω is any open set in Cd, and L is a suitable transfer operator acting on Bergman space A²(Ω), its eigenvalue sequence {λn(L)} is bounded by |λn(L)|?Aexp(−an1/d), where a,A>0 are explicitly given.     

Integral-type operators acting between weighted-type spaces on the unit ball

Let B denote the unit ball in Cⁿ and H(B) the space of all holomorphic functions on B. We study the boundedness and compactness of the following integral-type operators

     

Superposition operators between weighted spaces of analytic functions

Abstract

We study the existence and the continuity of superposition operators between weighted spaces of holomorphic functions in terms of the weights.     

Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W _p ^k (? ⁿ , E) with k ∈ ?₀, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.     

Characterizations of Disjointness preserving operators on vector-valued function spaces

We characterize compact and completely continuous disjointness preserving linear operators on vector-valued continuous functions as follows: a disjointness preserving operator is compact (resp. completely continuous) if and only if

for all

where is continuous and vanishes at infinity in the uniform (resp. strong) operator topology, and is compact (resp.  is uniformly completely continuous).