Essential Spectra

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Essential Spectrum ◽  
Constant Coefficient ◽  
Differential Operators ◽  
Adjoint Operator ◽  
Closed Operator ◽  
Essential Spectra

In this chapter, various essential spectra are studied. For a closed operator in a Banach space, a number of different sets have been used for the essential spectrum, the sets being identical for a self-adjoint operator in a Hilbert space. As well as the essential spectra, the changes that occur when the operator is perturbed are discussed. Constant-coefficient differential operators are studied in detail.

Discrete spectra criteria for singular differential operators with middle terms

1975 ◽  
Vol 77 (2) ◽  
pp. 337-347 ◽  
Author(s):  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Linear Operator ◽  
Differential Operators ◽  
Adjoint Operator ◽  
Continuous Functions ◽  
Adjoint Extension ◽  
Image Position ◽  
Discrete Spectra ◽  
Bounded Below

Let l be the differential operator of order 2n defined bywhere the coefficients are real continuous functions and pn > 0. The formally self-adjoint operator l determines a minimal closed symmetric linear operator L0 in the Hilbert space L2 (0, ∞) with domain dense in L2 (0, ∞) ((4), § 17). The operator L0 has a self-adjoint extension L which is not unique, but all such L have the same continuous spectrum ((4), § 19·4). We are concerned here with conditions on the pi which will imply that the spectrum of such an L is bounded below and discrete.

Essential Spectra of General Second-Order Differential Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Essential Spectrum ◽  
Distance Function ◽  
Differential Operators ◽  
Second Order ◽  
Compact Perturbation ◽  
Essential Spectra ◽  
Sectorial Operators

In this chapter, the operators considered are those m-sectorial operators discussed in Chapter VII, and the essential spectra are the sets defined in Chapter IX that remain invariant under compact perturbation. A generalization of a result of Persson is used to determine the least point of the essential spectrum. Davies’ mean distance function is introduced and consequences investigated.

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.

On the essential spectra of linear 2nth order differential operators with complex coefficients

1982 ◽  
Vol 92 (1-2) ◽  
pp. 65-75 ◽  
Author(s):  
David Race
Keyword(s):  
Differential Expression ◽  
Essential Spectrum ◽  
Differential Operators ◽  
Essential Spectra ◽  
Associated Operators ◽  
Linear Differential ◽  
Selfadjoint Extensions ◽  
Complex Valued ◽  
Complex Coefficients ◽  
Liouville Operators

SynopsisIn this paper, a formally J-symmetric, linear differential expression of 2nth order, with complex-valued coefficients, is considered. A number of results concerning the location of the essential spectrum of associated operators are obtained. These are extensions of earlier work dealing with complex Strum-Liouville operators, and include results which, in the real case, are due to Birman, Glazman and others. They lead to criteria, for the non-emptiness of the regularity field, of the corresponding minimal operator-a condition which is needed in the theory of J-selfadjoint extensions.

scholarly journals On a Class of Non-Self-Adjoint Differential Operators

1960 ◽  
Vol 12 ◽  
pp. 641-659 ◽  
Author(s):  
R. R. D. Kemp
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Spectral Analysis ◽  
Differential Operator ◽  
Spectral Measure ◽  
Differential Operators ◽  
Expansion Theorem ◽  
Completely Continuous ◽  
Normal Operators ◽  
Uniformly Bounded

The problem of spectral analysis of non-self-adjoint (and non-normal) operators has received considerable attention recently. Livšic (5), and more recently Brodskii and Livšic (1) have considered operators on Hilbert space with completely continuous imaginary parts. Dunford (3) has generalized the notion of spectral measure and defined a class of spectral operators on Hilbert and Banach space. Schwartz (8) and Rota (7) have investigated conditions under which a differential operator will be spectral. The work of Naimark (6) and the author (4) on non-self-adjoint differential operators leads to an expansion theorem which implicitly defines a type of spectral measure. However the projections involved in this will not in general be bounded, much less uniformly bounded.

scholarly journals Domains of elliptic operators on sets in Wiener space

2020 ◽  
Vol 23 (01) ◽  
pp. 2050004
Author(s):  
Davide Addona ◽  
Gianluca Cappa ◽  
Simone Ferrari
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Quadratic Form ◽  
Gaussian Measure ◽  
Adjoint Operator ◽  
The Self ◽  
Affine Function ◽  
Surface Measure ◽  
Regular Convex

Let [Formula: see text] be a separable Banach space endowed with a non-degenerate centered Gaussian measure [Formula: see text]. The associated Cameron–Martin space is denoted by [Formula: see text]. Consider two sufficiently regular convex functions [Formula: see text] and [Formula: see text]. We let [Formula: see text] and [Formula: see text]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form [Formula: see text] and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein–Uhlenbeck operator in Hilbert space with [Formula: see text] and on half-spaces, namely if [Formula: see text] and [Formula: see text] is an affine function, then the domain of the operator defined via (0.1) is the space [Formula: see text] where [Formula: see text] is the Feyel–de La Pradelle Hausdorff–Gauss surface measure.

scholarly journals Spectral radius of S-essential spectra

2020 ◽  
Vol 54 (1) ◽  
pp. 91-97
Author(s):  
C. Belabbaci
Keyword(s):  
Banach Space ◽  
Spectral Radius ◽  
Essential Spectrum ◽  
Measure Of Noncompactness ◽  
Dimensional Subspace ◽  
Linear Operators ◽  
Essential Spectra ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Kuratowski Measure Of Noncompactness

In this paper, we study the spectral radius of some S-essential spectra of a bounded linear operator defined on a Banach space. More precisely, via the concept of measure of noncompactness,we show that for any two bounded linear operators $T$ and $S$ with $S$ non zero and non compact operator the spectral radius of the S-Gustafson, S-Weidmann, S-Kato and S-Wolf essential spectra are given by the following inequalities\begin{equation}\dfrac{\beta(T)}{\alpha(S)}\leq r_{e, S}(T)\leq \dfrac{\alpha(T)}{\beta(S)},\end{equation}where $\alpha(.)$ stands for the Kuratowski measure of noncompactness and $\beta(.)$ is defined in [11].In the particular case when the index of the operator $S$ is equal to zero, we prove the last inequalities for the spectral radius of the S-Schechter essential spectrum. Also, we prove that the spectral radius of the S-Jeribi essential spectrum satisfies inequalities 2) when the Banach space $X$ has no reflexive infinite dimensional subspace and the index of the operator $S$ is equal to zero (the S-Jeribi essential spectrum, introduced in [7]as a generalisation of the Jeribi essential spectrum).

On an open problem of Weidmann: essential spectra and square-integrable solutions

2011 ◽  
Vol 141 (2) ◽  
pp. 417-430 ◽  
Author(s):  
Jiangang Qi ◽  
Shaozhu Chen
Keyword(s):  
Essential Spectrum ◽  
Open Problem ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Essential Spectra ◽  
Entire Interval ◽  
Linearly Independent ◽  
Adjoint Extension ◽  
Set Up ◽  
The Right

In 1987, Weidmann proved that, for a symmetric differential operator τ and a real λ, if there exist fewer square-integrable solutions of (τ−λ)y = 0 than needed and if there is a self-adjoint extension of τ such that λ is not its eigenvalue, then λ belongs to the essential spectrum of τ. However, he posed an open problem of whether the second condition is necessary and it has been conjectured that the second condition can be removed. In this paper, we first set up a formula of the dimensions of null spaces for a closed symmetric operator and its closed symmetric extension at a point outside the essential spectrum. We then establish a formula of the numbers of linearly independent square-integrable solutions on the left and the right subintervals, and on the entire interval for nth-order differential operators. The latter formula ascertains the above conjecture. These results are crucial in criteria of essential spectra in terms of the numbers of square-integrable solutions for real values of the spectral parameter.

AMD-Numbers, Compactness, Strict Singularity and the Essential Spectrum of Operators

2002 ◽  
Vol 9 (2) ◽  
pp. 227-270
Author(s):  
A. Castejón ◽  
E. Corbacho ◽  
V. Tarieladze
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Single Point ◽  
Minimal Elements ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Finite Dimensional ◽  
The Family

Abstract For an operator 𝑇 acting from an infinite-dimensional Hilbert space 𝐻 to a normed space 𝑌 we define the upper AMD-number and the lower AMD-number as the upper and the lower limit of the net (δ(𝑇|𝐸))𝐸∈𝐹𝐷(𝐻), with respect to the family 𝐹𝐷(𝐻) of all finite-dimensional subspaces of 𝐻. When these numbers are equal, the operator is called AMD-regular. It is shown that if an operator 𝑇 is compact, then and, conversely, this property implies the compactness of 𝑇 provided 𝑌 is of cotype 2, but without this requirement may not imply this. Moreover, it is shown that an operator 𝑇 has the property if and only if it is superstrictly singular. As a consequence, it is established that any superstrictly singular operator from a Hilbert space to a cotype 2 Banach space is compact. For an operator 𝑇, acting between Hilbert spaces, it is shown that and are respectively the maximal and the minimal elements of the essential spectrum of , and that 𝑇 is AMD-regular if and only if the essential spectrum of |𝑇| consists of a single point.

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