Friedrichs Extension and Min–Max Principle for Operators with a Gap

AbstractSome basics of a theory of unbounded Wiener–Hopf operators (WH) are developed. The alternative is shown that the domain of a WH is either zero or dense. The symbols for non-trivial WH are determined explicitly by an integrability property. WH are characterized by shift invariance. We study in detail WH with rational symbols showing that they are densely defined, closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are explicitly determined. Another topic concerns semibounded WH. There is a canonical representation of a semibounded WH using a product of a closable operator and its adjoint. The Friedrichs extension is obtained replacing the operator by its closure. The polar decomposition gives rise to a Hilbert space isomorphism relating a semibounded WH to a singular integral operator of Hilbert transformation type. This remarkable relationship, which allows to transfer results and methods reciprocally, is new also in the thoroughly studied case of bounded WH.

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On the Friedrichs extension of semi-bounded difference operators

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100072467 ◽

1994 ◽

Vol 116 (1) ◽

pp. 167-177 ◽

Cited By ~ 8

Author(s):

M. Benammar ◽

W. D. Evans

Keyword(s):

Liouville Equation ◽

Liouville Operator ◽

Difference Operators ◽

Dirichlet Integral ◽

Friedrichs Extension ◽

Sturm Liouville

In [5] Kalf obtained a characterization of the Friedrichs extension TF of a general semi-bounded Sturm–Liouville operator T, the only assumptions made on the coefficients being those necessary for T to be defined. The domain D(TF) of TF was described in terms of ‘weighted’ Dirichiet integrals involving the principal and non-principal solutions of an associated non-oscillatory Sturm–Liouville equation. Conditions which ensure that members of D(TF) have a finite Dirichlet integral were subsequently given by Rosenberger in [7].

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On the Right Hamiltonian for Singular Perturbations: General Theory

Reviews in Mathematical Physics ◽

10.1142/s0129055x97000221 ◽

1997 ◽

Vol 09 (05) ◽

pp. 609-633 ◽

Cited By ~ 3

Author(s):

Hagen Neidhardt ◽

Valentin Zagrebnov

Keyword(s):

General Theory ◽

Symmetric Operator ◽

Stability Domain ◽

Bounded Operators ◽

Friedrichs Extension ◽

Abstract Problem ◽

Dense Domain ◽

Adjoint Extension ◽

The Right ◽

Adjoint Operators

Let the pair of self-adjoint operators {A≥0,W≤0} be such that: (a) there is a dense domain [Formula: see text] such that [Formula: see text] is semibounded from below (stability domain), (b) the symmetric operator [Formula: see text] is not essentially self-adjoint (singularity of the perturbation), (c) the Friedrichs extension [Formula: see text] of [Formula: see text] is maximal with respect to W, i.e., [Formula: see text]. [Formula: see text]. Let [Formula: see text] be a regularizing sequence of bounded operators which tends in the strong resolvent sense to W. The abstract problem of the right Hamiltonian is: (i) to give conditions such that the limit H of self-adjoint regularized Hamiltonians [Formula: see text] exists and is unique for any self-adjoint extension [Formula: see text] of [Formula: see text], (ii) to describe the limit H. We show that under the conditions (a)–(c) there is a regularizing sequence [Formula: see text] such that [Formula: see text] tends in the strong resolvent sense to unique (right Hamiltonian) [Formula: see text], otherwise the limit is not unique.

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Self-Adjoint Extensions with Friedrichs Lower Bound

Complex Analysis and Operator Theory ◽

10.1007/s11785-020-01032-z ◽

2020 ◽

Vol 14 (7) ◽

Author(s):

Matteo Gallone ◽

Alessandro Michelangeli

Keyword(s):

Hilbert Space ◽

Lower Bound ◽

Symmetric Operator ◽

Simple Criterion ◽

Abstract Result ◽

Friedrichs Extension ◽

Lower Semi

Abstract We produce a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. Applications of this abstract result to a few instructive examples are then discussed.

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ON A FRIEDRICHS EXTENSION RELATED TO UNBOUNDED SUBNORMAL OPERATORS

Glasgow Mathematical Journal ◽

10.1017/s001708950500282x ◽

2006 ◽

Vol 48 (01) ◽

pp. 19 ◽

Cited By ~ 2

Author(s):

SAMEER CHAVAN ◽

AMEER ATHAVALE

Keyword(s):

Friedrichs Extension ◽

Subnormal Operators

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On the existence of nodal domains for elliptic differential operators

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500015651 ◽

1983 ◽

Vol 94 (3-4) ◽

pp. 287-299 ◽

Cited By ~ 4

Author(s):

E. Müller-Pfeiffer

Keyword(s):

Differential Equation ◽

Differential Operator ◽

Quadratic Form ◽

Unbounded Domain ◽

Dirichlet Boundary ◽

Differential Operators ◽

Dirichlet Boundary Conditions ◽

Friedrichs Extension ◽

Nodal Domains ◽

Elliptic Differential Operators

SynopsisWe prove that the selfadjoint elliptic differential equation (1) has rectangular nodal domains if the quadratic form of the equation takes on negative values. The existence of nodal domains is closely connected with the position of the smallest point of the spectrum of the corresponding selfadjoint operator (Friedrichs extension). If the smallest point of a second order selfadjoint differential operator with Dirichlet boundary conditions is an eigenvalue, then this eigenvalue is strictly increasing when the (possibly unbounded) domain, where the coefficients of the differential operator are denned, is shrinking (Theorem 4).

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