The spectra of Fredholm operators in locally convex spaces

Olagunju, P. A.; West, T. T.

doi:10.1017/s0305004100038275

Cited by 4 publications

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“…If the late P.A. Olagunju taught him the art of ghost writing [12], then he taught himself the art of spin, converting a simple error in the theory of bounded operators on incomplete normed spaces into a definition [15]. Trevor West has been an operator with single spectrum; this is not his obituary, but we do have [16] his epitaph: "here lies the West decomposition".…”

Section: Robin Hartementioning

confidence: 99%

Tim Starr, Trevor West and a positive contribution to operator theory

Harte

2007

Filomat

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Section: Robin Hartementioning

confidence: 99%

Tim Starr, Trevor West and a positive contribution to operator theory

Harte

2007

Filomat

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“…In both cases, the necessity of the condition is obvious. To prove the sufficiency, we note that, since tr {T*T} exists, } £ , j and the results follow from Lemmas 3.2 and 3.3 with tj = \Xj \ 2 . As the next theorem shows, these operators have a simple structure.…”

Section: If T Is a Compact Normal Operator In H The Following Conditmentioning

confidence: 99%

“…Remark. Provided that T satisfies certain conditions, this result may be given in the infinite dimensional case (see [2,Theorem 4.3]). …”

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“…Our basic result comes from [2]. A polynomial p will be called a complex polynomial if its coefficients are complex numbers; similarly for a real polynomial q. R will denote the real axis of the complex plane and R + the non-negative real axis.…”

mentioning

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“…A polynomial p will be called a complex polynomial if its coefficients are complex numbers; similarly for a real polynomial q. R will denote the real axis of the complex plane and R + the non-negative real axis. If n is a strictly positive integer, we shall denote the hypothesis " tr {T"(q(T) 2 } is real and non-negative for all real polynomials q " by the symbol Q(n, T). Remark.…”

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The spectra of compact operators in Hilbert spaces

West

1965

Proceedings of the Glasgow Mathematical Association

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In [2] a condition, originally due to Olagunju, was given for the spectra of certain compact operators to be on the real axis of the complex plane. Here, by using conformal mappings, this result is extended to more general curves. The problem divides naturally into two cases depending on whether or not the curve under consideration passes through the origin. Discussion is confined to the prototype curves C0 and C1. The case of C0, the unit circle of centre the origin, is considered in § 3; this problem is a simple one as the spectrum is a finite set. In § 4 results are given for C1 the unit circle of centre the point 1, and some results on ideals of compact operators, given in § 2, are needed. No attempt has been made to state results in complete generality (see [2]); this paper is kept within the framework of Hilbert space, and particularly simple conditions may be given if the operators are normal.

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A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions

Chiba

2015

Advances in Mathematics

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A spectral theory of linear operators on rigged Hilbert spaces (Gelfand triplets) is developed under the assumptions that a linear operator T on a Hilbert space H is a perturbation of a selfadjoint operator, and the spectral measure of the selfadjoint operator has an analytic continuation near the real axis in some sense. It is shown that there exists a dense subspace X of H such that the resolvent (λ − T ) −1 φ of the operator T has an analytic continuation from the lower half plane to the upper half plane as an X ′ -valued holomorphic function for any φ ∈ X, even when T has a continuous spectrum on R, where X ′ is a dual space of X. The rigged Hilbert space consists of three spaces X ⊂ H ⊂ X ′ . A generalized eigenvalue and a generalized eigenfunction in X ′ are defined by using the analytic continuation of the resolvent as an operator from X into X ′ . Other basic tools of the usual spectral theory, such as a spectrum, resolvent, Riesz projection and semigroup are also studied in terms of a rigged Hilbert space. They prove to have the same properties as those of the usual spectral theory. The results are applied to estimate asymptotic behavior of solutions of evolution equations.

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The spectra of Fredholm operators in locally convex spaces

Cited by 4 publications

References 4 publications

Tim Starr, Trevor West and a positive contribution to operator theory

Tim Starr, Trevor West and a positive contribution to operator theory

The spectra of compact operators in Hilbert spaces

A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions

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