Book ChapterDOI
M. Kreĭn’s Research on Semi-Bounded Operators, its Contemporary Developments, and Applications
Yu. Arlinskiĭ,E. Tsekanovskiĭ +1 more
- pp 65-112
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TLDR
In this paper, the authors consider the M Kreĭn classical papers on semi-bounded operators and the theory of contractive self-adjoint extensions of Hermitian contractions, and discuss their impact and role in the solution of J von Neumann's problem about parametrization in terms of his formulas of all nonnegative selfadjoint extension of nonnegative symmetric operators.Abstract:
We are going to consider the M Kreĭn classical papers on the theory of semi-bounded operators and the theory of contractive self-adjoint extensions of Hermitian contractions, and discuss their impact and role in the solution of J von Neumann’s problem about parametrization in terms of his formulas of all nonnegative self-adjoint extensions of nonnegative symmetric operators, in the solution of the Phillips-Kato extension problems (in restricted sense) about existence and parametrization of all proper sectorial (accretive) extensions of nonnegative operators, in bi-extension theory of non-negative operators with the exit into triplets of Hilbert spaces, in the theory of singular perturbations of nonnegative self-adjoint operators, in general realization problems (in system theory) of Stieltjes matrix-valued functions, in Nevanlinna-Pick system interpolation in the class of sectorial Stieltjes functions, in conservative systems theory with accretive main Schrodinger operator, in the theory of semi-bounded symmetric and self-adjoint operators invariant with respect to some groups of transformations New developments and applications to the singular differential operators are discussed as wellread more
Citations
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Journal ArticleDOI
On the Criteria of Transversality and Disjointness of Nonnegative Self-Adjoint Extensions of Nonnegative Symmetric Operators
TL;DR: In this article, a criterion of transversality and disjointness for the Friedrichs and Krein extensions of a nonnegative symmetric operator in terms of the vectors {φj, j ϵ 𝕁} that form a Riesz basis of the defect subspace was proposed.
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The Closed Extensions of a Closed Operator
TL;DR: In this paper, the graph norm of a densely defined and closed operator $A$ acting on a complex Hilbert space is characterized by a one-to-one correspondence between its closed extensions and subspaces that are closed with respect to graph norm.
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Extensions of dissipative operators with closable imaginary part
TL;DR: In this paper, a necessary and sufficient condition for an extension of a dissipative operator to still be dissipative was given, and the conditions for maximally accretive extensions of strictly positive symmetric operators and maximally dissipative extensions of a highly singular first-order operator on the interval were described.
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The Krein-von Neumann extension revisited
Guglielmo Fucci,Fritz Gesztesy,Klaus Kirsten,Klaus Kirsten,Lance L. Littlejohn,Roger Nichols,Jonathan Stanfill +6 more
TL;DR: In this paper, the Krein-von Neumann extension for singular, general Sturm-Liouville operators on arbitrary intervals is revisited and the boundary conditions are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.
Journal ArticleDOI
The Closed Extensions of a Closed Operator
TL;DR: In this article, a densely defined and closed (but not necessarily symmetric) operator A acting on a complex Hilbert space was studied, and a one-to-one correspondence between its closed extensions and subspaces that are closed with respect to the graph norm was established.
References
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Book
Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Book
Harmonic Analysis of Operators on Hilbert Space
TL;DR: In this article, the structure of operators of Class C 0.1 is discussed.Contractions and their Dilations, Geometrical and Spectral Properties of Dilations and Operator-Valued Analytic Functions are discussed.
Book
Solvable Models in Quantum Mechanics
TL;DR: The one-center point interaction as discussed by the authors is a special case of the Coulomb point interaction, where Coulomb plus one center point interaction in three dimensions plus Coulomb and one center interaction in two dimensions.
Journal ArticleDOI
Generalized resolvents and the boundary value problems for Hermitian operators with gaps
Vladimir Derkach,Mark Malamud +1 more
TL;DR: In this paper, a Hermitian operator A with gaps (αj, βj) (1 ⩽ j⩽ m ⩾ ∞) is studied and the self-adjoint extensions which put exactly kj < ∞ eigenvalues into each gap are described in terms of boundary conditions.
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