<p>The aim of this article is to detect the ascent and descent of weighted composition operators on Lorentz spaces. We investigate the conditions on a measurable transformation <i>T</i> and a complex-valued measurable function <i>u</i> defined on a measure space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((X,\mathcal {A},\mu )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="script">A</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> that cause the weighted composition operators on Lorentz space <i>L</i>(<i>p</i>,&#xa0;<i>q</i>), <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(1&lt;p\le \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(1\le q\le \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> to have finite or infinite ascent (descent). We also give a number of examples to illustrate our findings.</p>

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Ascent and descent of weighted composition operators on Lorentz spaces

  • Gopal Datt,
  • Daljeet Singh Bajaj

摘要

The aim of this article is to detect the ascent and descent of weighted composition operators on Lorentz spaces. We investigate the conditions on a measurable transformation T and a complex-valued measurable function u defined on a measure space \((X,\mathcal {A},\mu )\) ( X , A , μ ) that cause the weighted composition operators on Lorentz space L(pq), \(1<p\le \infty \) 1 < p , \(1\le q\le \infty \) 1 q to have finite or infinite ascent (descent). We also give a number of examples to illustrate our findings.