Weighted Composition Operators on Quasi-Banach Weighted Sequence Spaces

This paper studies the basic topological properties of weighted composition operators on the weighted sequence spaces \( l^{p}(\operatorname{w}) \), with \( 0<p<\infty \), given by a weight sequence \( \operatorname{w} \) of positive reals such as boundedness, compactness, compactness of differences of two operators, formulas for their essential norms, and description of closed range operators. Previously these properties were studied by Luan and Khoi in the case of Hilbert space \( (p=2) \). Their methods can be also applied with some minor modifications to the case of Banach spaces \( l^{p}(\operatorname{w}) \) with \( p>1 \). They based essentially on using the dual spaces of continuous linear functionals and, consequently, cannot be applied to the quasi-Banach case \( (0<p<1) \). Moreover, some of them do not work even in \( l^{1}(\operatorname{w}) \). Motivated by these reasons, we develop a more universal approach that allows studying the whole scale of spaces \( \{l^{p}(\operatorname{w}):p>0\} \). To this end, we establish the necessary and sufficient conditions for a linear operator to be compact on an abstract quasi-Banach sequence space. These conditions are new even in the case of Banach spaces. Moreover, we introduce the new characteristic, the \( \omega \)-essential norm of a continuous linear operator \( L \) on a quasi-Banach space \( X \). This characteristic measures the distance in the operator metric, between \( L \) and the set of all \( \omega \)-compact operators on \( X \). Here an operator \( K \) is \( \omega \)-compact on \( X \) if \( K \) is compact and coordinatewise continuous on \( X \). We show that for \( l^{p}(\operatorname{w}) \) with \( p>1 \) the essential and \( \omega \)-essential norms of a weighted composition operator coincide, whereas for \( 0<p\leq 1 \) we do not know whether the same is true or not. Our main results for weighted composition operators in \( l^{p}(\operatorname{w}) \) \( (0<p<\infty) \) are as follows: We provide criteria for an operator to be bounded, compact, or closed range, and completely describe the pairs of operators with compact difference; as well as some exact formula for the \( \omega \)-essential norm. Some key aspects of our approach can be used for other operators and scales of spaces.