Composition operators on classes of holomorphic functions on banach spaces

• Autores: Daniel Santacreu Ferrà
• Directores de la Tesis: Pablo Sevilla-Peris (dir. tes.) , David Jornet Casanova (dir. tes.)
• Lectura: En la Universitat Politècnica de València ( España ) en 2022
• Idioma: español
• Tribunal Calificador de la Tesis: Enrique Jorda Mora (presid.) , Manuel Domingo Contreras Márquez (secret.) , Quentin Menet (voc.)
• Enlaces
  • Tesis en acceso abierto en: RiuNet
• Resumen

  • The main aim in this thesis is to study different properties (mostly ergodic) of composition and weighted composition operators acting on spaces of holomorphic functions defined on an infinite dimensional complex Banach space.

    Let $X$ be a Banach space and $U$ some open subset. Given a mapping $\varphi \colon U \to U$ the action $f\mapsto C_{\varphi}(f)= f \circ \varphi$ defines an operator, called \emph{composition operator} (and $\varphi$ is called the \emph{symbol} of the operator). We consider this operator acting on different spaces of functions. The general philosophy is to try to characterise in each case the properties of our interest in terms of conditions on $\varphi$. Also, given $\psi\colon U \to \C$ the \emph{multiplication operator} is defined as $M_\psi(f)=\psi\cdot f$ and (with $\varphi$ as above), the \emph{weighted composition operator} as $C_{\psi,\varphi}(f)=\psi\cdot(f\circ \varphi)$ (here $\psi$ is called the \emph{weight} or \emph{multiplier} of the operator). Again, the idea is to describe properties of these operators in terms of conditions on $\psi$ and/or $\varphi$. Clearly $C_{\psi,\varphi}=M_\psi\circ C_\varphi$, and taking $\varphi=\id_U$ (the identity on $U$) or $\psi\equiv 1$ (the constant function $1$) we recover $M_\psi$ and $C_\varphi$.

    We denote the open unit ball of $X$ by $B$.

    The space of all holomorphic functions $f\colon B\to \C$ is denoted by $H(B)$. We write $H_b(B)$ for the space holomorphic functions of bounded type on $B$, and $H^\infty(B)$ for the space of bounded holomorphic functions on $B$. We are going to consider composition and weighted composition operators defined on each one of these spaces (taking then $U=B$ in the definition). We also consider composition operators defined on the vector space of all continuous $m$-homogeneous polynomials on $X$ (which is denoted by $\mathcal{P}(^{m} X)$). In this case we take $U=X$.

    The thesis consists of 5 chapters. In Chapter~\ref{General results} we introduce definitions and basic results, needed to make the text self-contained. In Chapter~\ref{Polynomials} we deal with mean ergodic and power bounded composition operators defined on $\mathcal{P}(^{m} X)$. In Chapter~\ref{ME Composition} we study mean ergodic and power bounded composition operators defined on $H(B)$, $H_b(B)$ and $H^\infty(B)$; considering also the particular case when $B$ is the ball of a Hilbert space. In Chapter~\ref{Compact Weighted Composition} we study compactness of weighted composition operators defined on $H^\infty(B)$, as well as boundedness, reflexivity, being Montel and (weak) compactness on $H_b(B)$. Finally, in Chapter~\ref{Mean ergodic Weighted Composition} we obtain different results about power boundedness and mean ergodicity of weighted composition operators acting on $H(B)$, $H_b(B)$ and $H^\infty(B)$, as well as about compactness and mean ergodicity of the multiplication operator.