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<p>We develop a Fredholm theory for continuous families of bounded linear operators on Krein spaces, and extend classical Hilbert space results to the indefinite inner product setting. We introduce the notions of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {J}\)</EquationSource> </InlineEquation>-index, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {J}\)</EquationSource> </InlineEquation>-Fredholm, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {J}\)</EquationSource> </InlineEquation>-Weyl, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {J}\)</EquationSource> </InlineEquation>-Browder families, together with their associated spectra. Within this framework, we establish <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathcal {J}\)</EquationSource> </InlineEquation>-Weyl- and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathcal {J}\)</EquationSource> </InlineEquation>-Browder-type theorems, prove spectral mapping results, and analyze stability under perturbations. Several examples are provided to illustrate the distinctions among these classes of operator families and demonstrate the applicability of the theory.</p>

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Fredholm theory of continuous families of bounded linear operators in Krein spaces

  • 期刊论文

  • 发表日期：

    2026年1月7日

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  • Hatem Baloudi,
  • Mohamed Ali Dbeibia,
  • Kamel Mahfoudhi

摘要

We develop a Fredholm theory for continuous families of bounded linear operators on Krein spaces, and extend classical Hilbert space results to the indefinite inner product setting. We introduce the notions of \(\mathcal {J}\) -index, \(\mathcal {J}\) -Fredholm, \(\mathcal {J}\) -Weyl, and \(\mathcal {J}\) -Browder families, together with their associated spectra. Within this framework, we establish \(\mathcal {J}\) -Weyl- and \(\mathcal {J}\) -Browder-type theorems, prove spectral mapping results, and analyze stability under perturbations. Several examples are provided to illustrate the distinctions among these classes of operator families and demonstrate the applicability of the theory.