##
**Weighted composition operators and dynamical systems on weighted spaces of holomorphic functions on Banach spaces.**
*(English)*
Zbl 1307.47021

Let \(X\) be a complex Banach space and let \(U_X\) be a balanced open subset of \(X\). Let \(V\) denote a Nachbin family of continuous weight functions \(v: U_X\to \mathbb{R^+}\) such that for every \(x\in U_X\) there exists \(v\in V\) for which \(v (x) > 0\). Let \(H (U_X)\) be the space of all holomorphic functions \(f: U_X \to \mathbb{C}\). The weighted spaces of holomorphic functions on \(U_X\) associated with V are defined as follows:
\[
HV(U_X)=\{f\in H (U_X): \|f\|_v=\sup_{x\in U_X}v(x)|f(x)|<\infty \text{ for all }v\in V\},
\]
and
\[
HV_0(U_X)=\{f\in HV (U_X): v \text{ vanishes at infinity outside } U_X\text{-bounded sets for all } v\in V\}.
\]
If \(V\) is a countable family of continuous weights, then \(HV (U_X)\) and \(HV_0 (U_X)\) are weighted Fréchet spaces. If the family of continuous weights has a unique element \(V = \{v\}\) such that \(v (x) > 0\) for all \(x\in U_X\), then \(HV (U_X)\) and \(HV_0 (U_X)\) endowed with the norm \(\|\cdot \|_v\) are Banach spaces.

In the paper under review, for fixed Banach spaces \(X\) and \(Y\), the author investigates the holomorphic mappings \(\phi: U_X\to U_Y\) and \(\psi : U_X\to\mathbb{C}\) for which the weighted composition operators \(W_{\phi,\psi}: H(U_Y)\to H(U_X)\), \(f\mapsto W_{\phi,\psi}f=\psi\cdot (f\circ\phi)\), is continuous between the spaces \(HV (U_X)\) (or \(HV_0 (U_X)\)) and \(H_W (U_Y )\) (or \(HW_0 (U_Y )\)). In particular, the continuity of \(W_{\phi,\psi}\) is characterized when \(U_X=B_X\) and \(U_Y=B_Y\), where \(B_X\) and \(B_Y\) denote the unit closed ball of \(X\) and \(Y\), respectively. Also, the author obtains a (linear) dynamical system induced by multiplication operators on these weighted spaces.

In the paper under review, for fixed Banach spaces \(X\) and \(Y\), the author investigates the holomorphic mappings \(\phi: U_X\to U_Y\) and \(\psi : U_X\to\mathbb{C}\) for which the weighted composition operators \(W_{\phi,\psi}: H(U_Y)\to H(U_X)\), \(f\mapsto W_{\phi,\psi}f=\psi\cdot (f\circ\phi)\), is continuous between the spaces \(HV (U_X)\) (or \(HV_0 (U_X)\)) and \(H_W (U_Y )\) (or \(HW_0 (U_Y )\)). In particular, the continuity of \(W_{\phi,\psi}\) is characterized when \(U_X=B_X\) and \(U_Y=B_Y\), where \(B_X\) and \(B_Y\) denote the unit closed ball of \(X\) and \(Y\), respectively. Also, the author obtains a (linear) dynamical system induced by multiplication operators on these weighted spaces.

Reviewer: Angela Albanese (Lecce)

### MSC:

47B33 | Linear composition operators |

47B38 | Linear operators on function spaces (general) |

47D03 | Groups and semigroups of linear operators |

37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |

32A10 | Holomorphic functions of several complex variables |

30H05 | Spaces of bounded analytic functions of one complex variable |