On the essential spectrum of self-adjoint operators

1980 ◽  
Vol 86 (3-4) ◽  
pp. 261-274
Author(s):  
B. J. Harris
Keyword(s):  
Differential Expression ◽  
Essential Spectrum ◽  
Adjoint Operator ◽  
Matrix Differential ◽  
Image Position ◽  
Adjoint Operators

SynopsisWe provide estimates of the formfor the length of gap centre μ in the essential spectrum of a self-adjoint operator generated by a matrix differential expression.

On the spectral analysis of self adjoint operators generated by second order difference equations

1991 ◽  
Vol 118 (1-2) ◽  
pp. 139-151 ◽  
Author(s):  
Dale T. Smith
Keyword(s):  
Essential Spectrum ◽  
Difference Equations ◽  
Difference Operator ◽  
Simple Criterion ◽  
Order Difference ◽  
Forward Difference ◽  
Other Information ◽  
Image Position ◽  
Adjoint Operators ◽  
Oscillation Constant

SynopsisIn this paper, I shall consider operators generated by difference equations of the formwhere Δ is the forward difference operator, and a, p, and r are sequences of real numbers. The connection between the oscillation constant of this equation and the bottom of the essential spectrum of self-adjoint extensions of the operator generated by the equation is given, as well as various other information about the spectrum of such extensions. In particular, I derive conditions for the spectrum to have only countably many eigenvalues below zero, and a simple criterion for the invariance of the essential spectrum.

Spectral Multipliers for Laplacians Associated with some Dirichlet Forms

2008 ◽  
Vol 51 (3) ◽  
pp. 581-607 ◽  
Author(s):  
Andrea Carbonaro ◽  
Giancarlo Mauceri ◽  
Stefano Meda
Keyword(s):  
Lebesgue Measure ◽  
Essential Spectrum ◽  
Dirichlet Form ◽  
Adjoint Operator ◽  
Dirichlet Forms ◽  
The Self ◽  
Spectral Multipliers ◽  
Curvature Condition ◽  
Image Position ◽  
Conditions At Infinity

AbstractLet be the self-adjoint operator associated with the Dirichlet formwhere ϕ is a positive C2 function, dλϕ = ϕdλ and λ denotes Lebesgue measure on ℝd. We study the boundedness on Lp(λϕ) of spectral multipliers of . We prove that if ϕ grows or decays at most exponentially at infinity and satisfies a suitable ‘curvature condition’, then functions which are bounded and holomorphic in the intersection of a parabolic region and a sector and satisfy Mihlin-type conditions at infinity are spectral multipliers of Lp(λϕ). The parabolic region depends on ϕ, on p and on the infimum of the essential spectrum of the operator on L2(λϕ). The sector depends on the angle of holomorphy of the semigroup generated by on Lp(λϕ).

Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)

1999 ◽  
Vol 129 (1) ◽  
pp. 165-179
Author(s):  
Yurii B. Orochko
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Differential Expression ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Deficiency Indices ◽  
Image Position ◽  
Symmetric Differential Operators

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expressionacting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

Spectral asymmetry and Riemannian geometry. III

1976 ◽  
Vol 79 (1) ◽  
pp. 71-99 ◽  
Author(s):  
M. F. Atiyah ◽  
V. K. Patodi ◽  
I. M. Singer
Keyword(s):  
Fundamental Group ◽  
Real Number ◽  
Analytic Continuation ◽  
Riemannian Geometry ◽  
Compact Manifold ◽  
Adjoint Operator ◽  
The Real ◽  
Spectral Asymmetry ◽  
Image Position ◽  
Adjoint Operators

In Parts I and II of this paper ((4), (5)) we studied the ‘spectral asymmetry’ of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator A on a compact manifold we definedwhere λ runs over the eigenvalues of A. For the particular operators of interest in Riemannian geometry we showed that ηA(s) had an analytic continuation to the whole complex s-plane, with simple poles, and that s = 0 was not a pole. The real number ηA(0), which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.

Monotone continuity of the spectral resolution and the eigenvalues

1980 ◽  
Vol 85 (1-2) ◽  
pp. 131-136 ◽  
Author(s):  
Joachim Weidmann
Keyword(s):  
Essential Spectrum ◽  
Spectral Resolution ◽  
Adjoint Operator ◽  
Resolvent Convergence ◽  
Adjoint Operators

SynopsisLet (Tn) be a non-decreasing sequence of self-adjoint operators which are semi-bounded from below and converge to some self-adjoint operator T in the sense of strong resolvent convergence. For every λ which is eventually below the essential spectrum of Tn it is shown that ‖En(λ)−E(λ)‖→0 for n→∞, where En(·) and E(·) are the spectral resolution of Tn and T, respectively.

On the almost sure approximation of self-adjoint operators in L2 (0, 1)

1996 ◽  
Vol 119 (3) ◽  
pp. 537-543
Author(s):  
L. J. Ciach ◽  
R. Jajte ◽  
A. Paszkiewicz
Keyword(s):  
Adjoint Operator ◽  
Almost Sure Convergence ◽  
The Other ◽  
Orthogonal Projections ◽  
Weak Operator Topology ◽  
Weak Operator ◽  
Image Position ◽  
Adjoint Operators ◽  
Orthogonal Systems ◽  
Martingale Convergence

There are several important theorems concerning the almost sure convergence of (monotone) sequences of orthogonal projections in L2-spaces. Let us mention here the martingale convergence theorems or the results on the developments of functions with respect to orthogonal systems. On the other hand every self-adjoint operator with the spectrum on the interval [0, 1] is a limit of some sequence of orthogonal projections in the weak operator topology (see [1]). This paper is devoted to a problem of approximation of a self-adjoint operator A acting in L2 (0, 1) by a sequence Pn of orthogonal projections in the sense that

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

18.—Asymptotic Estimates for the Lengths of the Gaps in the Essential Spectrum of Self-adjoint Differential Operators

1976 ◽  
Vol 74 ◽  
pp. 239-252 ◽  
Author(s):  
M. S. P. Eastham ◽  
W. N. Everitt
Keyword(s):  
Essential Spectrum ◽  
Differential Operators ◽  
Second Order ◽  
Asymptotic Estimates ◽  
Image Position ◽  
Differentiability Properties

SynopsisThe paper gives asymptotic estimates of the formas λ→∞ for the length l(μ)of a gap, centre μ in the essential spectrum associated with second-order singular differential operators. The integer r will be shown to depend on the differentiability properties of the coefficients in the operators and, in fact, r increases with the increasing differentiability of the coefficients. The results extend to all r ≧ – 2 the long-standing ones of Hartman and Putnam [10], who dealt with r = 0, 1, 2.

scholarly journals On the essential spectrum of Banach-space operators

2000 ◽  
Vol 43 (3) ◽  
pp. 511-528 ◽  
Author(s):  
Jörg Eschmeier
Keyword(s):  
Banach Space ◽  
Essential Spectrum ◽  
Negative Index ◽  
Resolvent Set ◽  
Image Position ◽  
Nuclear Spaces ◽  
Simply Connected ◽  
Banach Space Operators

AbstractLet T and S be quasisimilar operators on a Banach space X. A well-known result of Herrero shows that each component of the essential spectrum of T meets the essential spectrum of S. Herrero used that, for an n-multicyclic operator, the components of the essential resolvent set with maximal negative index are simply connected. We give new and conceptually simpler proofs for both of Herrero's results based on the observation that on the essential resolvent set of T the section spaces of the sheavesare complete nuclear spaces that are topologically dual to each other. Other concrete applications of this result are given.

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