Compact and Weakly Compact Homomorphisms on Fréchet Algebras of Holomorphic Functions

Abstract

We study homomorphisms between Fréchet algebras of holomorphic functions of bounded type. In this setting we prove that any pointwise bounded homomorphism into the space of entire functions of bounded type is rank one. We characterize up to the approximation property of the underlying Banach space, the weakly compact composition operators on Hb(V), V absolutely convex open set.

The Composition Operation on Spaces of Holomorphic Mappings

“…To realize this statement it suffices to recall that any mapping in H b (E, E) maps limited sets into limited sets ([13, Proposition 15]). It is a closed ideal because limited sets have the Grothendieck encapsulating property, i. e., a set L such that for every ε > 0 there is a limited set K ε ⊂ E such that L ⊂ B(0, ε) + K ε , is necessarily limited (see for instance [14,Lemma 2.4]).…”

Composition Operators on Classes of Holomorphic Functions on Banach Spaces

“…Composition operators in this setting have been much less studied, and the literature is more scarce. In [3,31] continuity, compactness and other related properties of these operators were studied. Let us collect the two results of those papers that are more related to our work.…”

“…Compactness of weighted composition operators defined on spaces of functions in have been extensively studied, and there is a huge related literature (see, for example, [10,17,18,19,30,52] and the references therein). However, for spaces of holomorphic functions on infinite dimensional spaces there are only a few references; for instance, the compactness of the composition operator C ϕ defined on H ∞ (B) was studied in [3] and on H b (B) in [31]. See Section 1.5 below for more detailed information about previous results.…”

Operadores de composição entre álgebras uniformes

“…Neste contexto, diversos resultados recentes sobre operadores de composição entre álgebras de funções analíticas podem ser encontrados nos trabalhos de P. Galindo [1, 7, 8, 12, 9, 11], R. Aron [1], M. Lindströn [1, 7, 8, 9, 11], R. Ryan [11], F. Behrouzi [2,3], T. W. Gamelin [7,8], H. Mahyar [3], L. Lourenço [12], L. Moraes [12], D. Carando [4]. Tais trabalhos dedicaram-se a estudar a propriedades como compacidade e compacidade fraca do operador de composição T g em termos da aplicação g. Em particular, em [1], os autores provam que um operador de composição T g : H ∞ (B E ) → H ∞ (B E ) é compacto se e somente se T g é fracamente compacto e g(B E ) é relativamente compacto em E, onde B E é a bola unitária do espaço de Banach E. Em [12], os autores mostram que todo operador de composição pontualmente limitado T g : A → H b (U ) possui imagem na álgebra de Banach H ∞ (U ). Estes dois resultados, entre outros, são generalizados nesta tese.…”

“…Lembramos que D(X) é a uB-álgebra de todas as funções complexas definidas e limitadas em X, com a norma||f || = sup x∈X |f (x)|, ∀f ∈ D(X) (Veja Seção 2.1). Em particular, temos que D(X) ⊂ B b (X).A demonstração da Proposição 2.51 abaixo segue as mesmas idéias da Proposição 2.1 de[12].…”