On Compressions of Self-Adjoint Extensions of a Symmetric Linear Relation with Unequal Deficiency Indices
- Resource Type
- Original Paper
- Authors
- Mogilevskii, Vadim I.
- Source
- Journal of Mathematical Sciences: A Translation of Selected Russian- and Ukrainian-language Serial Publications in Mathematics. 246(5):671-686
- Subject
- Symmetric and self-adjoint linear relation (operator)
exit space self-adjoint extension
compression
boundary triplet
- Language
- English
- ISSN
- 1072-3374
1573-8795
Let A be a symmetric linear relation in the Hilbert space ℌ with unequal deficiency indices n−A < n+(A). A self-adjoint linear relation A˜⊃Aℌ˜⊃ℌCA˜=PℌA˜↾ℌA˜=A∗˜.CA˜A˜=A∗˜.CA˜A˜=A∗˜.in some Hilbert space A˜⊃Aℌ˜⊃ℌCA˜=PℌA˜↾ℌA˜=A∗˜.CA˜A˜=A∗˜.CA˜A˜=A∗˜.is called an (exit space) extension of A. We study the compressions A˜⊃Aℌ˜⊃ℌCA˜=PℌA˜↾ℌA˜=A∗˜.CA˜A˜=A∗˜.CA˜A˜=A∗˜. of extensions A˜⊃Aℌ˜⊃ℌCA˜=PℌA˜↾ℌA˜=A∗˜.CA˜A˜=A∗˜.CA˜A˜=A∗˜. Our main result is a description of compressions A˜⊃Aℌ˜⊃ℌCA˜=PℌA˜↾ℌA˜=A∗˜.CA˜A˜=A∗˜.CA˜A˜=A∗˜. by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter τ(λ) from the Krein formula for generalized resolvents. We describe also all extensions A˜⊃Aℌ˜⊃ℌCA˜=PℌA˜↾ℌA˜=A∗˜.CA˜A˜=A∗˜.CA˜A˜=A∗˜. of A with the maximal symmetric compression A˜⊃Aℌ˜⊃ℌCA˜=PℌA˜↾ℌA˜=A∗˜.CA˜A˜=A∗˜.CA˜A˜=A∗˜. and all extensions A˜⊃Aℌ˜⊃ℌCA˜=PℌA˜↾ℌA˜=A∗˜.CA˜A˜=A∗˜.CA˜A˜=A∗˜. of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators A with equal deficiency indices n+(A) = n−(A).