Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\left( \Omega ,\Sigma ,m\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>m</mi> </mfenced> </math></EquationSource> </InlineEquation> be a measure space with m being <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-finite positive measure and let T be a spectral operator on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^{p}\left( \Omega \right) :=L^{p}\left( \Omega ,\Sigma ,m\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mfenced close=")" open="("> <mi mathvariant="normal">Ω</mi> </mfenced> <mo>:</mo> <mo>=</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mfenced close=")" open="("> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>m</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\left( 1\le p<\infty \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mfenced> </math></EquationSource> </InlineEquation> space. We study convergence (in various topologies) of the sequences <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\left\{ T^{n}f\right\} _{n\in \mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mfenced close="}" open="{"> <msup> <mi>T</mi> <mi>n</mi> </msup> <mi>f</mi> </mfenced> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L^{p}\left( \Omega \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mfenced close=")" open="("> <mi mathvariant="normal">Ω</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> spaces. Some results concerning ergodic properties of spectral operators are given. Some related problems are also discussed.

Convergence of Iterates of Spectral Operators in $L^{p}$-Spaces

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发表日期：

2026年7月27日

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Heybetkulu S. Mustafayev

摘要

Let $\left( \Omega ,\Sigma ,m\right) $ $(Ω, Σ, m)$ be a measure space with m being $\sigma $ $σ$ -finite positive measure and let T be a spectral operator on $L^{p}\left( \Omega \right) :=L^{p}\left( \Omega ,\Sigma ,m\right) $ $L^{p} (Ω) : = L^{p} (Ω, Σ, m)$ $\left( 1\le p<\infty \right) $ $(1 \leq p < \infty)$ space. We study convergence (in various topologies) of the sequences $\left\{ T^{n}f\right\} _{n\in \mathbb {N}}$ ${\{T^{n}, f\}}_{n \in N}$ in $L^{p}\left( \Omega \right) $ $L^{p} (Ω)$ spaces. Some results concerning ergodic properties of spectral operators are given. Some related problems are also discussed.

Convergence of Iterates of Spectral Operators in \(L^{p}\)-Spaces

摘要