In this paper we study dominated orthogonally additive (in general nonlinear) operators on Köthe–Bochner spaces. We resolve the open problem stated in [<CitationRef CitationID="CR12">12</CitationRef>], showing that there exist Köthe–Bochner spaces E(X), F(Y) and an orthogonally additive operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$T:E(X)\rightarrow F(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, such that T is f-locally dominated for each <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f\in E(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, but is not dominated. We also prove that there is a C-bounded orthogonally additive operator <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T:E(X)\rightarrow F(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> which is not strongly C-bounded, resolving the open problem stated in [<CitationRef CitationID="CR25">25</CitationRef>].

Dominated Operators on Köthe–Bochner Spaces

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发表日期：

2026年2月24日

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Alina Gutnova,
Vladimir Lapushkin,
Marat Pliev,
Fedor Sukochev

摘要

In this paper we study dominated orthogonally additive (in general nonlinear) operators on Köthe–Bochner spaces. We resolve the open problem stated in [12], showing that there exist Köthe–Bochner spaces E(X), F(Y) and an orthogonally additive operator $T:E(X)\rightarrow F(Y)$ $T : E (X) \to F (Y)$ , such that T is f-locally dominated for each $f\in E(X)$ $f \in E (X)$ , but is not dominated. We also prove that there is a C-bounded orthogonally additive operator $T:E(X)\rightarrow F(Y)$ $T : E (X) \to F (Y)$ which is not strongly C-bounded, resolving the open problem stated in [25].