Generalized Resolvents of an lsometric Operator in a Pontrjagin Space

Everitt has shown [1[, that for α ∊ [0, π/2] the undernoted problem (1.1–2) with an indefinite weight function r can be represented by a selfadjoint operator in a suitable Hilbert space. This result is extended to arbitrary α ∊ [0, π), replacing the Hilbert space in some cases by a Pontrjagin space with index one. The problem is also treated in the Krein space generated by the weight function r.

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A Pontrjagin space analysis of the supercritical transport equation

Transport Theory and Statistical Physics ◽

10.1080/00411457508204840 ◽

1975 ◽

Vol 4 (4) ◽

pp. 143-154 ◽

Cited By ~ 6

Author(s):

Joseph A. Ball ◽

William Greenberg

Keyword(s):

Transport Equation ◽

Space Analysis ◽

Pontrjagin Space

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On generalized resolvents of symmetric linear relations in a Pontrjagin space

Annales Academiae Scientiarum Fennicae Series A I Mathematica ◽

10.5186/aasfm.1978-79.0425 ◽

1979 ◽

Vol 4 ◽

pp. 365-370

Author(s):

Pekka Sorjonen

Keyword(s):

Linear Relations ◽

Pontrjagin Space

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Definitizable Operators on a Krein Space

Canadian Mathematical Bulletin ◽

10.4153/cmb-1995-072-7 ◽

1995 ◽

Vol 38 (4) ◽

pp. 496-506 ◽

Cited By ~ 1

Author(s):

Petr Zizler

Keyword(s):

Linear Operator ◽

Selfadjoint Operator ◽

Inner Product ◽

Indefinite Inner Product ◽

Bounded Linear ◽

Positive Type ◽

Approximate Point Spectrum ◽

Definitizable Operators ◽

Pontrjagin Space ◽

Operator Pencils

AbstractLet A be a bounded linear operator on a Hilbert space H. Assume that A is selfadjoint in the indefinite inner product defined by a selfadjoint, bounded, invertible linear operator G on H; [x,y] := (Gx,y). In the first part of the paper we define two orders of neutrality for the pair (G, A) and a connection is made with the "types" of numbers in the point and approximate point spectrum of A. The main results of the paper are in the second part and they deal with strong and uniform definitizability of a bounded selfadjoint operator on a Pontrjagin space. They state:A) Let A be a bounded strongly definitizable operator on a Pontrjagin space ΠK, then A is uniformly definitizable.B) A bounded selfadjoint operator A on a Pontrjagin space ΠK is uniformly definitizable if and only if all the eigenvalues of A are of definite type and all the nonisolated eigenvalues of A are of positive type.Some applications to the theory of linear selfadjoint operator pencils are given.

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