Weighted Composition Operators on Quasi-Banach Weighted Sequence Spaces

Abanin, A. V. , Mannanikov, R. S.
Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 4.

Abstract:
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.

Keywords: quasi-Banach sequence spaces, weighted composition operators, weighted \(l^p\) spaces

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