Truncation method for operators with disconnected essential spectrum

Namboodiri, M. N. N.

doi:10.1007/bf02829650

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…Also here we are able to compute one end point of a gap. The other end point is possible to compute by Theorem 2.3 of [18], which is stated below. Coming back to the Arveson's class, we observe that the essential points and hence the essential spectrum is fully determined by the zeros of the function in the Definition 3.3.…”

Section: Definition 33mentioning

confidence: 99%

Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

Kumar¹

2015

Semigroups, Algebras and Operator Theory

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Normal categories are essentially those arising as the category of principal left [right] ideals of a regular semigroup. These categories have been used in describing the structure of regular semigroups. The structure theory in this context is known as cross connection theory. Several associated categories can be derived from a normal category which are also of interest in the structure theory of regular semigroups. The subcategory of inclusions, the subcategory of retractons, the groupoid of isomorphisms etc. are some of the associated categories.

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Section: Definition 33mentioning

confidence: 99%

Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

Kumar¹

2015

Semigroups, Algebras and Operator Theory

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“…The spectral gap issues of such operators were studied with the linear algebraic techniques in [6]. The spectral gap issues of arbitrary bounded self-adjoint operators can be found in the literature (see [7,8] for example).…”

Section: Discussionmentioning

confidence: 99%

Essential Spectrum of Discrete Laplacian – Revisited

Kumar¹

2022

3C TIC

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Consider the discrete Laplacian operator A acting on l2(Z). It is well known from the classical literature that the essential spectrum of A is a compact interval. In this article, we give an elementary proof for this result, using the finite-dimensional truncations An of A. We do not rely on symbol analysis or any infinite-dimensional arguments. Instead, we consider the eigenvalue-sequences of the truncations An and make use of the filtration techniques due to Arveson. Usage of such techniques to the discrete Schrödinger operator and to the multi-dimensional settings will be interesting future problems.

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Perturbation of operators and approximation of spectrum

2014

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Let A (x) be a norm continuous family of bounded self-adjoint operators on a separable Hilbert space H and let A(x)(n) be the orthogonal compressions of A (x) to the span of first n elements of an orthonormal basis of H. The problem considered here is to approximate the spectrum of A(x) using the sequence of eigenvalues of A(x)(n). We show that the bounds of the essential spectrum and the discrete spectral values outside the bounds of essential spectrum of A(x) can be approximated uniformly on all compact subsets by the sequence of eigenvalue functions of A(x)(n). The known results, for a bounded self-adjoint operator, are translated into the case of a norm continuous family of operators. Also an attempt is made to predict the existence of spectral gaps that may occur between the bounds of essential spectrum of A(0) = A and study the effect of norm continuous perturbation of operators in the prediction of spectral gaps. As an example, gap issues of some block Toeplitz-Laurent operators are discussed. The pure linear algebraic approach is the main advantage of the results here

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Truncation method for operators with disconnected essential spectrum

Cited by 3 publications

References 6 publications

Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

Essential Spectrum of Discrete Laplacian – Revisited

Perturbation of operators and approximation of spectrum

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