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 lp(w), 0 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 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 ω-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 ω-compact operators on X. Here an operator K is called ω-compact on X if it is compact and coordinate-wise continuous on X. In this relation it is shown that for lp(w) with p > 1 the essential and ω-essential norms of a weighted composition operator coincide while for 0 function show_eabstract() { $('#eabstract1').hide(); $('#eabstract2').show(); $('#eabstract_expand').hide(); }