Weighted composition operators on quasi-Banach weighted sequence spaces

This paper is devoted to the basic topological properties of weighted composition operators on the weighted sequence spaces $l^p(\text{w})$, $0<p<\infty$, given by a weight sequence $\text{w}$ of positive numbers such as boundedness, compactness, compactness of differences of two operators, formulas for their essential norms, and a description of those operators that have a closed range. Previously these properties were studied by D. M. Luan and L. H. 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 $l^p(\text{w})$, $p>1$. They are essentially based on the use of conjugate spaces of linear continuous functionals and, consequently, cannot be applied to the quasi-Banach case $(0<p<1)$. Moreover, some of them do not work even in the Banach space $l^1(\text{w})$. Motivated by these reasons we develop a more universal approach that allows to study the whole scale $\{l^p(\text{w}) : p>0 \}$. To do this we establish necessary and sufficient conditions for a linear operator to be compact on an abstract quasi-Banach sequence space which are new also for the case of Banach spaces. In addition it is introduced a new characteristic which is called $\omega$-essential norm of a linear continuous operator $L$ on a quasi-Banach space $X$. It measures the distance, in operator metric, between $L$ and the set of all $\omega$-compact operators on $X$. Here an operator $K$ is called $\omega$-compact on $X$ if it is compact and coordinate-wise continuous on $X$. In this relation it is shown that for $l^p(\text{w})$ with $p>1$ the essential and $\omega$-essential norms of a weighted composition operator coincide while for $0 < p \le 1$ we do not know whether the same result is true or not. Our main results for weighted composition operators on $l^p(\text{w})$ $(0 < p <\infty)$ are the following: criteria for an operator to be bounded, compact, or have a closed range; a complete description of pairs of operator with compact difference; an exact formula for $\omega$-essential norm. Some key aspects of our approach can be used for other operators and scales of spaces.