{"title":"Weighted Composition Operators on Quasi-Banach Weighted Sequence Spaces","authors":"A. V. Abanin, R. S. Mannanikov","doi":"10.1134/s0037446624040165","DOIUrl":null,"url":null,"abstract":"<p>This paper studies the basic topological properties of weighted composition operators\non the weighted sequence spaces <span>\\( l^{p}(\\operatorname{w}) \\)</span>, with <span>\\( 0<p<\\infty \\)</span>,\ngiven by a weight sequence <span>\\( \\operatorname{w} \\)</span> of positive reals\nsuch as boundedness, compactness, compactness of differences of two operators, formulas for their essential norms,\nand description of closed range operators.\nPreviously these properties were studied by Luan and Khoi in the case of Hilbert space <span>\\( (p=2) \\)</span>.\nTheir methods can be also applied with some minor modifications to the case of Banach spaces <span>\\( l^{p}(\\operatorname{w}) \\)</span>\nwith <span>\\( p>1 \\)</span>.\nThey based essentially on using the dual spaces of continuous linear functionals and, consequently,\ncannot be applied to the quasi-Banach case <span>\\( (0<p<1) \\)</span>. Moreover, some of them do not work even\nin <span>\\( l^{1}(\\operatorname{w}) \\)</span>.\nMotivated by these reasons, we develop a more universal approach that allows studying\nthe whole scale of spaces <span>\\( \\{l^{p}(\\operatorname{w}):p>0\\} \\)</span>.\nTo this end, we establish the necessary and sufficient conditions for a linear operator to be compact\non an abstract quasi-Banach sequence space. These conditions are new even in the case of Banach spaces.\nMoreover, we introduce the new characteristic, the <span>\\( \\omega \\)</span>-essential norm of a continuous linear operator <span>\\( L \\)</span>\non a quasi-Banach space <span>\\( X \\)</span>.\nThis characteristic measures the distance in the operator metric, between <span>\\( L \\)</span> and the set of all <span>\\( \\omega \\)</span>-compact operators on <span>\\( X \\)</span>.\nHere an operator <span>\\( K \\)</span> is <span>\\( \\omega \\)</span>-compact on <span>\\( X \\)</span> if <span>\\( K \\)</span> is compact and coordinatewise continuous on <span>\\( X \\)</span>.\nWe show that for <span>\\( l^{p}(\\operatorname{w}) \\)</span> with <span>\\( p>1 \\)</span> the essential and <span>\\( \\omega \\)</span>-essential norms\nof a weighted composition operator coincide, whereas for <span>\\( 0<p\\leq 1 \\)</span> we do not know whether the same is true or not.\nOur main results for weighted composition operators in <span>\\( l^{p}(\\operatorname{w}) \\)</span> <span>\\( (0<p<\\infty) \\)</span> are as follows:\nWe provide criteria for an operator to be bounded, compact, or closed range, and completely describe the pairs of operators\nwith compact difference; as well as some exact formula for the <span>\\( \\omega \\)</span>-essential norm. Some key aspects of\nour approach can be used for other operators\nand scales of spaces.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0037446624040165","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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.
本文研究了加权序列空间 \( l^{p}(\operatorname{w}) \)上加权组成算子的基本拓扑性质,其中 \( 0<p<\infty \)由正实数的加权序列 \( \operatorname{w} \)给出,这些性质包括有界性、紧凑性、两个算子差的紧凑性、它们的本质规范公式以及闭区间算子的描述。他们的方法在稍作修改后也可以应用于巴纳赫空间(l^{p}(\operatorname{w}) \)与(p>1 \)的情况。他们基本上基于使用连续线性函数的对偶空间,因此不能应用于准巴纳赫情况((0<p<1) \)。基于这些原因,我们开发了一种更通用的方法,可以研究整个尺度的空间。\为此,我们建立了线性算子在抽象准巴纳赫序列空间上紧凑的必要条件和充分条件。此外,我们还引入了一个新的特性--准巴纳赫空间(X)上连续线性算子(L)的基本规范(essential norm of a continuous linear operator \( L \)on a quasi-Banach space \( X \).这个特性度量了算子度量中,(L)与(X)上所有紧凑算子的集合(the set of all \( \omega \)-compact operators on \( X \)之间的距离。这里,如果一个算子(K)在(X)上是紧凑且坐标连续的,那么这个算子(K)在(X)上就是紧凑的。我们证明,对于有 p>1 的 \( l^{p}(\operatorname{w}) \),加权合成算子的本质规范和本质规范是重合的,而对于 \( 0<p\leq 1 \),我们不知道这是否是真的。\(0<p<\infty) \)中的加权组成算子的主要结果如下:我们提供了算子有界、紧凑或闭合范围的标准,并完整地描述了具有紧凑差分的算子对;以及一些关于 \(\omega \)-本质规范的精确公式。我们方法的一些关键方面可用于其他算子和空间尺度。