Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866910888839086080 |
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| author | Borogovac, Muhamed |
| author_facet | Borogovac, Muhamed |
| contents | Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function $Q$, represented by a self-adjoint linear relation $A$, is decomposed by means of the reducing subspaces of $A$. The sum of two functions $Q_{i}{\in N}_{κ_{i}}\left( \mathcal{H} \right),\thinspace i=1,\thinspace 2$, minimally represented by the triplets $\left( \mathcal{K}_{i},A_{i},Γ_{i} \right)$, is also studied. For that purpose, a model $( \tilde{\mathcal{K}},\tilde{A},\tilde{Γ} )$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $\left( \mathcal{K}_{i},A_{i},Γ_{i} \right)$ is created. By means of that model, necessary and sufficient conditions for $κ=κ_{1}+κ_{2}$ are proven in analytic terms. At the end, it is explained how degenerate Jordan chains of the representing relation $A$ affect reducing subspaces of $A$ and decomposition of the corresponding function $Q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_00725 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions Borogovac, Muhamed Functional Analysis (2010) } 46C20 47A06 47B50 33E99 Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function $Q$, represented by a self-adjoint linear relation $A$, is decomposed by means of the reducing subspaces of $A$. The sum of two functions $Q_{i}{\in N}_{κ_{i}}\left( \mathcal{H} \right),\thinspace i=1,\thinspace 2$, minimally represented by the triplets $\left( \mathcal{K}_{i},A_{i},Γ_{i} \right)$, is also studied. For that purpose, a model $( \tilde{\mathcal{K}},\tilde{A},\tilde{Γ} )$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $\left( \mathcal{K}_{i},A_{i},Γ_{i} \right)$ is created. By means of that model, necessary and sufficient conditions for $κ=κ_{1}+κ_{2}$ are proven in analytic terms. At the end, it is explained how degenerate Jordan chains of the representing relation $A$ affect reducing subspaces of $A$ and decomposition of the corresponding function $Q$. |
| title | Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions |
| topic | Functional Analysis (2010) } 46C20 47A06 47B50 33E99 |
| url | https://arxiv.org/abs/2010.00725 |