On composition operators between weighted (LF)‐ and (PLB)‐spaces of continuous functions

Angela A. Albanese and Claudio Mele

Mathematische Nachrichten, 2023, vol. 296, issue 12, 5384-5399

Abstract: Let X be a locally compact Hausdorff topological space, let V=(vn,k)n,k∈N$\mathcal {V}=(v_{n,k})_{n,k\in {\mathbb {N}}}$ be a system of positive continuous functions on X and let φ be a continuous self‐map on X. The composition operators Cφ:f↦f∘φ$C_\varphi : f\mapsto \ f\,\circ\, \varphi$ on the weighted function (LF)‐spaces VC(X)$\mathcal {V}C(X)$ (V0C(X)$\mathcal {V}_0C(X)$, resp.) and on the weighted function (PLB)‐spaces AC(X)$\mathcal {A}C(X)$ (A0C(X)$\mathcal {A}_0C(X)$, resp.) are studied. We characterize when the operator Cφ$C_\varphi$ acts continuously on such spaces in terms of the system V$\mathcal {V}$ and the map φ, as well as we determine conditions on V$\mathcal {V}$ and φ which correspond to various basic properties of the composition operator Cφ$C_\varphi$, like boundedness, compactness, and weak compactness. Our approach requires a study of the continuity, boundedness, (weak) compactness of the linear operators between (LF)‐spaces and (PLB)‐spaces.

Date: 2023
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