scholarly journals On joint essential spectra of doubly commuting n-tuples of p-hyponormal operators

1998 ◽  
Vol 40 (3) ◽  
pp. 353-358 ◽  
Author(s):  
In Ho Jeon
Keyword(s):  
Essential Spectrum ◽  
Polar Decomposition ◽  
Essential Spectra ◽  
Weyl Spectrum ◽  
Hyponormal Operators ◽  
Image Position

AbstractLet A be an operator on a Hillbert space with polar decomposition A = |A|, let  = |A|½U|A|½ and let  = V|Â| be the polar decomposition of Â. Write à for the operatorà = |Â|½V|Â|½. If = (A1,…,AN) is a doubly commuting n-tuple of p-hyponormal operators on a Hillbert space with equal defect and nullity, then = (Ã1,…,Ãn) is a doubly commuting n-tuple of hyponormal operators. In this paper we show thatwhere σ* denotes σTe (Taylor essential spectrum), σTw (Taylor-Weyl spectrum) and σTb (Taylor-Browder spectrum), respectively.

scholarly journals A remark on the essential spectra of quasi-similar dominant contractions

1989 ◽  
Vol 31 (2) ◽  
pp. 165-168
Author(s):  
B. P. Duggal
Keyword(s):  
Complex Hilbert Space ◽  
Normal Extension ◽  
Inclusion Relation ◽  
Linear Transformations ◽  
Essential Spectra ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Hyponormal Operators ◽  
Image Position ◽  
Dominant Operator

We consider operators, i.e. bounded linear transformations, on an infinite dimensional separable complex Hilbert space H into itself. The operator A is said to be dominant if for each complex number λ there exists a number Mλ(≥l) such that ∥(A – λ)*x∥ ≤ Mλ∥A – λ)x∥ for each x∈H. If there exists a number M≥Mλ for all λ, then the dominant operator A is said to be M-hyponormal. The class of dominant (and JW-hyponormal) operators was introduced by J. G. Stampfli during the seventies, and has since been considered in a number of papers, amongst then [7], [11]. It is clear that a 1-hyponormal is hyponormal. The operator A*A is said to be quasi-normal if Acommutes with A*A, and we say that A is subnormal if A has a normal extension. It is known that the classes consisting of these operators satisfy the following strict inclusion relation:

scholarly journals Weyl's theorem holds for p-hyponormal operators

1997 ◽  
Vol 39 (2) ◽  
pp. 217-220 ◽  
Author(s):  
Muneo Chō ◽  
Masuo Itoh ◽  
Satoru Ōshiro
Keyword(s):  
Point Spectrum ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear ◽  
Approximate Point Spectrum ◽  
Weyl Spectrum ◽  
Hyponormal Operators ◽  
Image Position ◽  
Isolated Eigenvalues

Let ℋ be a complex Hilbert space and B(ℋ) the algebra of all bounded linear operators on ℋ. Let ℋ(ℋ) be the algebra of all compact operators of B(ℋ). For an operator T ε B(ℋ), let σ(T), σp(T), σπ(T) and πoo(T) denote the spectrum, the point spectrum, the approximate point spectrum and the set of all isolated eigenvalues of finite multiplicity of T, respectively. We denote the kernel and the range of an operator T by ker(T) and R(T), respectively. For a subset of ℋ, the norm closure of is denoted by . The Weyl spectrum ω(T) of T ε B(ℋ) is defined as the set

scholarly journals Lifting sets and the Calkin algebra

1982 ◽  
Vol 23 (1) ◽  
pp. 83-84 ◽  
Author(s):  
G. J. Murphy
Keyword(s):  
Hilbert Space ◽  
Essential Spectrum ◽  
Compact Operators ◽  
Linear Operators ◽  
Infinite Dimension ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Weyl Spectrum ◽  
Image Position ◽  
The Ideal

H will denote a Hilbert space of infinite dimension, ℬ(H) the algebra of bounded linear operators on H, and ℛ(H) the ideal of compact operators on H. We let σ, σe and σω denote the spectrum, essential spectrum and Weyl spectrum respectively. It is well known that for arbitrary T ∈ ℬ(H) we have by [5]andand

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.

On the spectra of non-self-adjoint realisations of second-order elliptic operators

1981 ◽  
Vol 90 (1-2) ◽  
pp. 71-105 ◽  
Author(s):  
W. D. Evans
Keyword(s):  
Spherical Shell ◽  
Essential Spectrum ◽  
Weighted Space ◽  
Elliptic Operators ◽  
Second Order ◽  
Sectorial Operator ◽  
Image Position ◽  
Second Order Elliptic Operators ◽  
Complex Valued

SynopsisLet τ denote the second-order elliptic expressionwhere the coefficients bj and q are complex-valued, and let Ω be a spherical shell Ω = {x:x ∈ ℝn, l <|x|<m} with l≧0, m≦∞. Under the conditions assumed on the coefficients of τ and with either Dirichlet or Neumann conditions on the boundary of Ω, τ generates a quasi-m-sectorial operator T in the weighted space L2(Ω;w). The main objective is to locate the spectrum and essential spectrum of T. Best possible results are obtained.

scholarly journals Factorization of hyponormal operators

1972 ◽  
Vol 13 (3) ◽  
pp. 323-326
Author(s):  
Mary R. Embry
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Hyponormal Operators ◽  
Image Position ◽  
Continuous Linear Operators

In [1]R. G. Douglas proved that if A and B are continuous linear operators on a Hilbert space X, the following three conditions are equivalent:

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.

The least limit point of the spectrum associated with sinǵular differential operators

1970 ◽  
Vol 67 (2) ◽  
pp. 277-281 ◽  
Author(s):  
M. S. P. Eastham
Keyword(s):  
Differential Operator ◽  
Linear Operator ◽  
Essential Spectrum ◽  
Limit Point ◽  
Compact Support ◽  
Complex Space ◽  
Differential Operators ◽  
Adjoint Extension ◽  
Image Position

Let τ be the formally self-adjoint differential operator denned bywhere the pr(x) are real-valued, , and p0(x) > 0. Then τ determines a real symmetric linear operator T0, given by T0f = τf, whose domain D(T0) consists of those functions f in the complex space L2(0, ∞) which have compact support and 2n continuous derivatives in (0, ∞) and vanish in some right neighbourhood of x = 0 ((7), p. 27–8). Since D(T0) is dense in L2(0, ∞), T0 has a self-adjoint extension T. We denote by μ the least limit point of the spectrum of T. The operator T may not be unique, but all such T have the same essential spectrum ((7), p. 28) and therefore μ does not depend on the choice of T.

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.

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