Dynamics of weighted composition operators on Stein manifolds

Duality Theory for Spaces of Germs and Holomorphic Functions on Nuclear Spaces

In this paper, we present certain results on convolution equations and shift-invariant subspaces in special spaces of entire functions of exponential type. Problems related to a convolution equation and invariant subspaces were considered in various spaces of holomorphic functions (see, e.g., the surveys [1, 3] and the books [4, 5]), and the choice of the given spaces of holomorphic functions is related to the fact that in many other spaces of holomorphic functions, there arise difficulties connected with the harmonic analysis and synthesis, i.e., with the existence of invariant subspaces in which, as was proved by D. I. Gurevich [10], either there are no polynomial–exponential element, or they are not dense in a subspace. In the chosen class, there are none of these initial difficulties. 1. Space of Entire Functions PG Let G be a Runge domain in the space C n , O(G) be the space of functions holomorphic in G equipped with the natural topology, i.e., the topology of the uniform convergence on all compact sets in G. Denote by O � (G) the dual space to O(G), i.e., the space of all linear continuous functionals on O(G )e quipped with the strong topology. Sometimes, elements of T ∈O � (G) are called analytic functionals on the domain G. The space O(G) is a closed subspace of the space C 0 (G) of all functions continuous in G, and, therefore, for every analytic functional T ∈O � (G), there exists a measure µ, supp µ G, such that

We consider holomorphic functions $\phi$ taking the unit disc $U$ into itself, and Banach spaces $X$ consisting of functions holomorphic in $U$ and continuous on its closure; and show that under some natural hypotheses on $X$: if $\phi$ induces a compact composition operator on $X$, then $\phi (U)$ must be a relatively compact subset of $U$. Spaces $X$ which satisfy the hypotheses of this theorem include the disc algebra, "heavily" weighted Dirichlet spaces, spaces of holomorphic Lipschitz functions, and the space of functions with derivative in a Hardy space ${H^p}(p \geq 1)$. It is well known that the theorem is not true for "large" spaces such as the Hardy and Bergman spaces. Surprisingly, it also fails in "very small spaces," such as the Hilbert space of holomorphic functions $f(z) = \sum {{a_n}{z^n}}$ determined by the condition $\sum {|{a_n}{|^2}\exp \left ( {\sqrt n } \right ) < \infty }$. The property of Möbiusinvariance plays a crucial and mysterious role in these matters.

Weighted spaces of harmonic and holomorphic functions on the unit disc are studied. We show that for all radial weights which are not decreasing too fast the space of harmonic functions is isomorphic to c0. For the weights that we consider we completely characterize those spaces of holomorphic functions which are isomorphic to c0. Moreover, we determine when the Riesz projection, mapping the weighted space of harmonic functions onto the corresponding space of holomorphic functions, is bounded.

Let D be a Cartan domain in Cd and let G = Aut(D) be the group of all biholomorphic automorphisms of G. Consider the projective representation of G on spaces of holomorphic functions on D ((Uv(g)f)(z) {j(g-1))((z))) {j(g -1)(z)}(z)(1())g∈G where z is the genus of D and W is in the Wallach set D. We identify the minimal and the maximal Uv ((G))-invariant Banach spaces of holomorphic functions on D in a very explicit way: The minimal space 𝔐v is a Besov-1 space, and the maximal space Mv is a weighted 8-space. Moreover, with respect to the pairing under the (unique) U(v)(Uv)- invariant inner product we have 𝔐v G =Mv. In the second part of the paper we consider invariant Banach spaces of vector-valued holomorphic functions and obtain analogous descriptions of the unique maximal and minimal space, in particular for the important special case of “constant” partitions which arises naturally in connection with nontube type domains.

Weakly uniformly continuous holomorphic functions and the approximation property

In this paper we study the tensor representation of spaces of Frechet-valued holomorphic functions \([H(U, F),\tau ]\) in the form \( [(H(U), \tau )] \widehat{\otimes }_{\pi }F\) where U is an open subset of a Frechet space and \(\tau \in \{\tau _0, \tau _\omega , \tau _\delta \}.\) Using this result we consider the following problems: exponential laws for the topologies \(\tau _0, \tau _\omega \) on the space \(H(U \times V)\) where U and V are two open subsets of locally convex spaces E and F respectively; the coincidence of the topologies \(\tau _0, \tau _\omega , \tau _\delta \) on spaces of locally convex valued holomorphic functions (resp. germs) H(U, F) [resp. H(K, F)]; the inheritance of the properties (QNo), \((QNo)'\) via the spaces of holomorphic functions and of holomorphic germs.

Duality and Spaces of Holomorphic Functions

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in n -dimensional Euclidean space \mathbb R^n . It is proved that if n\ge 2 , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, and associative if and only if it is L_p addition for some 1\le p\le\infty . It is also demonstrated that if n\ge 2 , an operation * between compact convex sets is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, and has the identity property (i.e., K*\{o\}=K=\{o\}*K for all compact convex sets K , where o denotes the origin) if and only if it is Minkowski addition. Some analogous results for operations between star sets are obtained. Various characterizations are given of operations that are projection covariant, meaning that the operation can take place before or after projection onto subspaces, with the same effect. Several other new lines of investigation are followed. A relatively little-known but seminal operation called M -addition is generalized and systematically studied for the first time. Geometric-analytic formulas that characterize continuous and \operatorname{GL}(n) -covariant operations between compact convex sets in terms of M -addition are established. It is shown that if n\ge 2 , an o -symmetrization of compact convex sets (i.e., a map from the compact convex sets to the origin-symmetric compact convex sets) is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, and translation invariant if and only if it is of the form \lambda DK for some \lambda\ge 0 , where DK=K+(-K) is the difference body of K . The term “polynomial volume” is introduced for the property of operations * between compact convex or star sets that the volume of rK*sL , r,s\ge 0 , is a polynomial in the variables r and s . It is proved that if n\ge 2 , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, associative, and has polynomial volume if and only if it is Minkowski addition.

We study some nonstandard spaces of functions holomorphic in domains on the complex plain with certain smoothness conditions up to the boundary. The first type is the space of Holder-type holomorphic functions with prescribed modulus of continuity $$\omega =\omega (h)$$ , and the second is the variable exponent holomorphic Holder space with the modulus of continuity $$|h|^{\lambda (z)}$$ . We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.

In this chapter we apply the methods and results of Oka theory to a variety of problems in Stein geometry. In particular, we discuss the structure of holomorphic vector bundles and their homomorphisms over Stein spaces, find the minimal number of generators of coherent analytic sheaves over Stein spaces, consider the problem of complete intersections and, more generally, of elimination of intersections of holomorphic maps with complex subvarieties, present the solution of the holomorphic Vaserstein problem, discuss transversality theorems for holomorphic and algebraic maps from Stein manifolds, and obtain estimates on the dimension of singularity (degeneration) sets of generic holomorphic maps from Stein manifolds to Oka manifolds. In the last part of the chapter we further develop the method of local holomorphic sprays of maps from strongly pseudoconvex Stein domains. We apply this technique to prove an up-to-the-boundary version of the Oka principle on such domains, and we establish a Banach manifold structure theorem for certain spaces of holomorphic maps from strongly pseudoconvex domains to arbitrary complex manifolds.

Suppose that Y is a complex manifold such that any holomorphic map from a compact convex set in a Euclidean space ℂ n to Y is a uniform limit of entire maps ℂ n →Y. We prove that a holomorphic map X 0 →Y from a closed complex subvariety X 0 in a Stein manifold X admits a holomorphic extension X→Y provided that it admits a continuous extension. We then establish the equivalence of four Oka-type properties of a complex manifold.

The aim of this thesis is to study the ergodic properties of some operators defined on several spaces of functions. In a locally convex Hausdorff space E, an operator T\in L(E) is called power bounded if the set of its iterates is equicontinuous. The Cesaro means of T are T_[n] = 1/n (T+T^2+...+ T^m), n\in\N. The operator T is called mean ergodic if the sequence (T_[n])_n converges pointwise and it is called uniformly mean ergodic if the sequence converges uniformly on bounded sets. In Chapter 1, the multiplication operator is studied when defined on weighted spaces of continuous functions and their inductive and projective limits. We work with a Hausdorff, normal, locally compact topological space X. Given a continuous function phi (a symbol), the multiplication operator is M_ phi: f -> phi f. A continuous function v is a weight if it is strictly positive. The (Banach) weighted spaces of continuous functions are C_v:= {f\in C(X) : ||f||_v:=\sup_(x\in X) v(x)|f(x)|< infty}, C_v ^0 :={f\in C(X) : vf vanishes at infinity}, with the norm ||.||_v. The Sections 1.3 and 1.4 are devoted to inductive and projective limits of the spaces in Section 1.2. If V=(v_n)_n is a decreasing family of weights, the weighted inductive limits of continuous functions are VC=ind _n C_v_n and V_0C=ind _n C^0_v_n. If A=(a_n)_n is an increasing family of weights, the weighted projective limits of continuous functions are CA=proj_n C_a_n and CA_0=proj _n C^0_a_n. The behaviour is different for the limits of the C_v_n (resp. C_a_n) and the limits of the C^0_v_n (resp. C^0_a_n). In Section 1.5 the spectrum and the Waelbroeck spectrum are completely determined. In the final Section 1.6 the topology of the set of multipliers between projective limits is compared with the one induced by the operator topology of uniform convergence on bounded sets. The work of Chapter 2 is devoted to weighted sequence spaces and their inductive and projective limits. A sequence v=(v(i))_i \in \C^\N is called a weight if it is strictly positive. The weighted Banach spaces of sequences considered are l_p(v), 1<= p<= infty and c_0(v). Given A=(a_n)_n, a Kothe matrix, the echelon space of order 1<= p<= infty is defined by proj _n l _p (a_n) and proj _n c_0 (a_n). The co-echelon space of order 1<= p<= infty is defined, for a decreasing family of weights V=(v_n)_n, by ind_n l _p (v_n) and ind_n c_0 (v_n). In the Sections 2.2 and 2.3 ergodic and spectral properties of the multiplication operator are studied. In Section 2.4 it is characterized when the multiplication operator is bounded or compact, in similar terms than continuity. In Section 2.5, as in Section 1.6, the topology of the set of multipliers between echelon spaces is compared with the one induced by the operator topology of uniform convergence on bounded sets. Also the topology of the set of bounded multiplication operators is studied. In the final Section 2.6, the results of the previous sections are applied to the power series spaces, as particular cases of echelon spaces. Chapter 3 deals with the composition operator given by a holomorphic self-map of the complex open unit disc, when considered between different Banach spaces of holomorphic functions. If phi : \D - > \D is holomorphic, the composition operator is C_phi: f ->f o phi. In Section 3.2 necessary and sufficient conditions are given for ergodic properties of a composition operator defined on a general Banach space of holomorphic functions under the assumption of one or many of given properties. The results of Section 3.2 are applied in Section 3.3 to classical spaces of holomorphic functions, particularly, weighted Bergman spaces of infinite type H_v and H_v^0, Bloch spaces B_p and B_p ^0, Bergman spaces A^p and Hardy spaces H^p.

The authors solve an interesting open problem concerning the equivalence of the compact-open topology τ0 and the Nachbin ported topology τω on spaces of holomorphic functions. (See, for example, the book by S. Dineen [Complex analysis in locally convex spaces, North-Holland, Amsterdam, 1981; MR0640093 (84b:46050)] for background.) Let H(U) denote the space of complex-valued holomorphic functions on an open subset U of a complex Frechet-Montel space F. Ansemil and S. Ponte [Arch. Math. (Basel) 51 (1988), no. 1, 65–70; MR0954070 (90a:46109)] showed that these two topologies agree on H(U) for balanced U if and only if, for every natural number n, P(nF) is a Montel space. Using this result, they showed that for balanced open subsets U of certain non-Schwartz, Frechet-Montel spaces, τ0=τω. Earlier, J. Mujica [J. Funct. Anal. 57 (1984), no. 1, 31–48; MR0744918 (86c:46050)] had shown that τ0=τω for Frechet-Schwartz spaces. It is not hard to see that the two topologies differ if F is not Montel. The authors' counterexample is the Frechet-Montel space F of Taskinen [Studia Math. 91 (1988), no. 1, 17–30; MR0957282 (89k:46087)]. The authors observe that the complete symmetric projective tensor product Fs⊗ˆπF contains an isomorphic copy of l1. Consequently, P(2F) cannot be Montel, and the result follows.

The Ph.D. Thesis ¿Operators on weighted spaces of holomorphic functions¿ presented&#13;\nhere treats different areas of functional analysis such as spaces of holomorphic&#13;\nfunctions, infinite dimensional holomorphy and dynamics of operators.&#13;\nAfter a first chapter that introduces the notation, definitions and the basic results&#13;\nwe will use throughout the thesis, the text is divided into two parts. A first one,&#13;\nconsisting of Chapters 1 and 2, focused on a study of weighted (LB)-spaces of entire&#13;\nfunctions on Banach spaces, and a second one, corresponding to Chapters 3 and&#13;\n4, where we consider differentiation and integration operators acting on different&#13;\nclasses of weighted spaces of entire functions to study its dynamical behaviour. In&#13;\nwhat follows, we give a brief description of the different chapters:&#13;\nIn Chapter 1, given a decreasing sequence of continuous radial weights on a Banach&#13;\nspace X, we consider the weighted inductive limits of spaces of entire functions&#13;\nVH(X) and VH0(X). Weighted spaces of holomorphic functions appear naturally&#13;\nin the study of growth conditions of holomorphic functions and have been investigated&#13;\nby many authors since the work of Williams in 1967, Rubel and Shields&#13;\nin 1970 and Shields and Williams in 1971. We determine conditions on the family&#13;\nof weights to ensure that the corresponding weighted space is an algebra or&#13;\nhas polynomial Schauder decompositions. We study Hörmander algebras of entire&#13;\nfunctions defined on a Banach space and we give a description of them in terms of&#13;\nsequence spaces. We also focus on algebra homomorphisms between these spaces&#13;\nand obtain a Banach-Stone type theorem for a particular decreasing family of&#13;\nweights. Finally, we study the spectra of these weighted algebras, endowing them&#13;\nwith an analytic structure, and we prove that each function f ¿ VH(X) extends&#13;\nnaturally to an analytic function defined on the spectrum. Given an algebra homomorphism,&#13;\nwe also investigate how the mapping induced between the spectra&#13;\nacts on the corresponding analytic structures and we show how in this setting&#13;\ncomposition operators have a different behavior from that for holomorphic functions&#13;\nof bounded type. This research is related to recent work by Carando, García,&#13;\nMaestre and Sevilla-Peris. The results included in this chapter are published by&#13;\nBeltrán in [14]. Chapter 2 is devoted to study the predual of VH(X) in order to linearize this space&#13;\nof entire functions. We apply Mujica¿s completeness theorem for (LB)-spaces to&#13;\nfind a predual and to prove that VH(X) is regular and complete. We also study&#13;\nconditions to ensure that the equality VH0(X) = VH(X) holds. At this point,&#13;\nwe will see some differences between the finite and the infinite dimensional cases.&#13;\nFinally, we give conditions which ensure that a function f defined in a subset&#13;\nA of X, with values in another Banach space E, and admitting certain weak&#13;\nextensions in a space of holomorphic functions can be holomorphically extended&#13;\nin the corresponding space of vector-valued functions. Most of the results obtained&#13;\nhave been published by the author in [13].&#13;\nThe rest of the thesis is devoted to study the dynamical behaviour of the following&#13;\nthree operators on weighted spaces of entire functions: the differentiation operator&#13;\nDf(z) = f (z), the integration operator Jf(z) = z&#13;\n0 f(¿)d¿ and the Hardy&#13;\noperator Hf(z) = 1&#13;\nz z&#13;\n0 f(¿)d¿, z ¿ C.&#13;\nIn Chapter 3 we focus on the dynamics of these operators on a wide class of&#13;\nweighted Banach spaces of entire functions defined by means of integrals and&#13;\nsupremum norms: the weighted spaces of entire functions Bp,q(v), 1 ¿ p ¿ ¿,&#13;\nand 1 ¿ q ¿ ¿. For q = ¿ they are known as generalized weighted Bergman&#13;\nspaces of entire functions, denoted by Hv(C) and H0&#13;\nv (C) if, in addition, p = ¿.&#13;\nWe analyze when they are hypercyclic, chaotic, power bounded, mean ergodic&#13;\nor uniformly mean ergodic; thus complementing also work by Bonet and Ricker&#13;\nabout mean ergodic multiplication operators. Moreover, for weights satisfying&#13;\nsome conditions, we estimate the norm of the operators and study their spectrum.&#13;\nSpecial emphasis is made on exponential weights. The content of this chapter is&#13;\npublished in [17] and [15].&#13;\nFor differential operators ¿(D) : Bp,q(v) ¿ Bp,q(v), whenever D : Bp,q(v) ¿&#13;\nBp,q(v) is continuous and ¿ is an entire function, we study hypercyclicity and&#13;\nchaos. The chapter ends with an example provided by A. Peris of a hypercyclic&#13;\nand uniformly mean ergodic operator. To our knowledge, this is the first example&#13;\nof an operator with these two properties. We thank him for giving us permission&#13;\nto include it in our thesis.&#13;\nThe last chapter is devoted to the study of the dynamics of the differentiation and&#13;\nthe integration operators on weighted inductive and projective limits of spaces of&#13;\nentire functions. We give sufficient conditions so that D and J are continuous on&#13;\nthese spaces and we characterize when the differentiation operator is hypercyclic,&#13;\ntopologically mixing or chaotic on projective limits. Finally, the dynamics of these&#13;\noperators is investigated in the Hörmander algebras Ap(C) and A0&#13;\np(C). The results&#13;\nconcerning this topic are included by Bonet, Fernández and the author in [16].