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• W4307788783 abstract "UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function <mml:math> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> represented by a self-adjoint linear relation <mml:math> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> in a Pontryagin space is decomposed by means of the reducing subspaces of <mml:math> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> The sum of two functions <mml:math> <mml:mrow> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mrow> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>κ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo form=prefix>(</mml:mo> <mml:mi>ℋ</mml:mi> <mml:mo form=postfix>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> <mml:math> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1,2</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> minimally represented by the triplets <mml:math> <mml:mrow> <mml:mo form=prefix>(</mml:mo> <mml:msub> <mml:mi>𝒦</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo form=postfix>)</mml:mo> </mml:mrow> </mml:math> is also studied. For this purpose, we create a model <mml:math> <mml:mrow> <mml:mo form=prefix>(</mml:mo> <mml:mover accent=true> <mml:mi>𝒦</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mover accent=true> <mml:mi>A</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mover accent=true> <mml:mi>Γ</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo form=postfix>)</mml:mo> </mml:mrow> </mml:math> to represent <mml:math> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> in terms of <mml:math> <mml:mrow> <mml:mo form=prefix>(</mml:mo> <mml:msub> <mml:mi>𝒦</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo form=postfix>)</mml:mo> </mml:mrow> </mml:math>. By using this model, necessary and sufficient conditions for <mml:math> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation <mml:math> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> affect the reducing subspaces of <mml:math> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> and the decomposition of the corresponding function <mml:math> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>" @default.
• W4307788783 created "2022-11-05" @default.
• W4307788783 creator A5087042156 @default.
• W4307788783 date "2022-08-09" @default.
• W4307788783 modified "2023-10-18" @default.
• W4307788783 title "Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions" @default.
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• W4307788783 doi "https://doi.org/10.37863/umzh.v74i7.6084" @default.
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