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<p>This paper establishes novel characterizations of the Radon-Nikodým property (RNP) in Banach lattices through the lens of positive linear operators and vector measure theory. By employing operator-theoretic approaches, we demonstrate that a Banach lattice <i>E</i> possesses RNP if and only if every positive Dunford-Pettis operator from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_1[0,1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> into <i>E</i> admits Bochner integral representation. Furthermore, we investigate the implications of these results for evolution equations in ordered spaces, proving that the existence of non-constant strong solutions to autonomous equations governed by m-accretive operators fundamentally characterizes RNP in Banach lattices. Our findings refine classical results on differentiability of vector-valued functions and provide new insights into the structural properties of Banach lattices.</p>

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Radon-Nikodým property and strong solutions of evolution equations in banach lattices

  • 期刊论文

  • 发表日期：

    2026年8月3日

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  • Fu Zhang,
  • Zhen Ni,
  • Haixia Sun

摘要

This paper establishes novel characterizations of the Radon-Nikodým property (RNP) in Banach lattices through the lens of positive linear operators and vector measure theory. By employing operator-theoretic approaches, we demonstrate that a Banach lattice E possesses RNP if and only if every positive Dunford-Pettis operator from \(L_1[0,1]\) L 1 [ 0 , 1 ] into E admits Bochner integral representation. Furthermore, we investigate the implications of these results for evolution equations in ordered spaces, proving that the existence of non-constant strong solutions to autonomous equations governed by m-accretive operators fundamentally characterizes RNP in Banach lattices. Our findings refine classical results on differentiability of vector-valued functions and provide new insights into the structural properties of Banach lattices.