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Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.46698/x505725003053t Weighted Composition Operators on QuasiBanach Weighted Sequence Spaces
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 quasiBanach 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 quasiBanach 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 quasiBanach 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 coordinatewise 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: quasiBanach sequence spaces, weighted composition operators, weighted \(l^p\) spaces
Language: Russian
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For citation: Abanin, A. V. and Mannanikov, R. S. Weighted Composition Operators on QuasiBanach Weighted Sequence Spaces, Vladikavkaz Math. J., 2023, vol. 25, no. 4, pp. 519 (in Russian).
DOI 10.46698/x505725003053t ← Contents of issue 
 

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