Weighted composition operators on the predual and dual Banach spaces of Beurling algebras on \(\mathbb Z\) (Q1931566)

Let \(v\) be a \textit{weight sequence} on \(\mathbb{Z}\), i.e., a function \(\mathbb{Z}\overset{v}{\rightarrow}[1,\infty)\) for which \[ v( m+n) \leq v( m) v( n) \text{ for all }m,n\in\mathbb{Z}. \] Consider the Banach spaces \[ \ell^{\infty}( \mathbb{Z},1/v ) =\left\{ \mathbb{Z}\overset {f}{\rightarrow}\mathbb{C}:\sup_{n\in\mathbb{Z}}|f(n)| /v( n) <\infty\right\} \] and \[ c_{0}( \mathbb{Z},1/v) =\left\{ \mathbb{Z}\overset {f}{\rightarrow}\mathbb{C}:\lim_{|n|\rightarrow\infty }\left| f( n) \right| /v( n) <\infty \right\} \] with the natural norm \(\left\| f\right\| =\sup_{n\in\mathbb{Z} }\left| f( n) \right| /v( n) \). As a matter of fact, \(\ell^{\infty}( \mathbb{Z},1/v) \) (respectively, \(c_{0}( \mathbb{Z},1/v) \)) is the dual (respectively, predual) space of the space \[ \ell^{1}( \mathbb{Z},v) =\left\{ \mathbb{Z}\overset {f}{\rightarrow}\mathbb{C}:\sum_{n\in\mathbb{Z}}v( n) \left| f( n) \right| <\infty\right\} . \] Actually, \(\ell^{1}( \mathbb{Z},v) \) is a unital commutative Banach algebra with respect to the norm \(\left\| f\right\| =\sum _{n\in\mathbb{Z}}v( n) \left| f( n) \right| \) and the convolution product as multiplication. This interesting paper deals with weighted composition operators (in a usual sense) on both \(\ell^{\infty }( \mathbb{Z},1/v) \) and \(c_{0}( \mathbb{Z},1/v) \). Special attention is paid to boundedness and (weak) compactness properties.