This paper presents a comprehensive study of generalized Kato decompositions and Browder-type properties for linear relations in Banach spaces. We establish several characterizations of these classes through the existence of bounded projections that commute with the relation in a generalized sense. Specifically, we show that a linear relation T belongs to the generalized Kato class <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {GKD}(M,N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">GKD</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if there exists a bounded projection P such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$TP = PT + T(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mi>P</mi> <mo>=</mo> <mi>P</mi> <mi>T</mi> <mo>+</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T(0) \subset \ker (P)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>⊂</mo> <mo>ker</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T+P$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>+</mo> <mi>P</mi> </mrow> </math></EquationSource> </InlineEquation> is semi-regular, and PT is quasi-nilpotent. We extend these results to various subclasses including Kato decomposable relations, semi-Fredholm relations, essentially semi-regular relations, and generalized essentially Kato decomposable relations. Furthermore, we provide characterizations of left and right Browder relations through commuting projections with additional finite-dimensionality and invertibility properties. Our results unify and extend the classical theory of linear operators to the more general framework of linear relations.

Generalized Kato and Browder linear relations via commuting projections

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发表日期：

2026年4月24日

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Nihel Feki,
Maher Mnif

摘要

This paper presents a comprehensive study of generalized Kato decompositions and Browder-type properties for linear relations in Banach spaces. We establish several characterizations of these classes through the existence of bounded projections that commute with the relation in a generalized sense. Specifically, we show that a linear relation T belongs to the generalized Kato class $\mathcal {GKD}(M,N)$ $GKD (M, N)$ if and only if there exists a bounded projection P such that $TP = PT + T(0)$ $T P = P T + T (0)$ , $T(0) \subset \ker (P)$ $T (0) \subset ker (P)$ , $T+P$ $T + P$ is semi-regular, and PT is quasi-nilpotent. We extend these results to various subclasses including Kato decomposable relations, semi-Fredholm relations, essentially semi-regular relations, and generalized essentially Kato decomposable relations. Furthermore, we provide characterizations of left and right Browder relations through commuting projections with additional finite-dimensionality and invertibility properties. Our results unify and extend the classical theory of linear operators to the more general framework of linear relations.