Essential norm resolvent estimates and essential numerical range

Let B(H) denote the algebra of bounded linear operators on a complex Hilbert space H. The (classical) numerical range of T ∈ B(H) is the set W(T) = {〈T x; x〉: x ∈ H; ‖x‖ = 1} Writing T= T_1 + iT_2 for self-adjoint T_1, T_2 ∈ B(H), W(T) can be identified with the set {(〈T_1 x, x〉,〈T_2 x, x〉) : x ∈ H, ‖x‖ = 1}. This leads to the notion of the joint numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈ 〖B(H)〗^n. It is defined by W(T) = {(〈T_1 x, x〉,〈T_2 x, x〉, …, 〈T_n x, x〉) : x ∈ H, ‖x‖ = 1}. The joint numerical range has been studied extensively in order to understand the joint behaviour of operators. Let K(H) be the set of all compact operators on a Hilbert space H. The essential numerical range of T ∈ B(H) is defined by W_(e ) (T)=∩{W(T+K) :K∈K(H) }. The joint essential numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈〖 B(H)〗^n is defined analogously by W_(e ) (T)=∩{ /W(T+K) :K∈〖K(H)〗^n }. These notions have been generalized to operators on a Banach space. In Chapter 1 of this thesis, the joint spatial essential numerical range were introduced. Also the notions of the joint algebraic numerical range V(T) and the joint algebraic essential numerical range Ve(T) were reviewed. Basic properties of these sets were given. In 2010, Muller proved that each n-tuple of operators T on a separable Hilbert space has a compact perturbation T + K so that We(T) = W(T + K). In Chapter 2, it was shown that any n-tuple T of operators on lp has a compact perturbation T +K so that Ve(T) = V (T +K), provided that Ve(T) has an interior point. A key step was to find for each n-tuple of operators on lp a compact perturbation and a sequence of finite-dimensional subspaces with respect to which it is block 3 diagonal. This idea was inspired by a similar construction of Chui, Legg, Smith and Ward in 1979. Let H and L be separable Hilbert spaces and consider the operator D_AB=A⨂I_L⨂B on the tensor product space H ⨂▒L. In 1987 Magajna proved that W_(e ) (D_AB )=co[W_(e ) (A)- /(W(B)))∪/W(A) - W_(e ) (B))] by considering quasidiagonal operators. An alternative proof of the equality was given in Chapter 3 using block 3 diagonal operators. The maximal numerical range and the essential maximal numerical range of T ∈ B(H) were introduced by Stampi in 1970 and Fong in 1979 respectively. In 1993, Khan extended the notions to the joint essential maximal numerical range. However the set may be empty for some T ∈ B(H). In Chapter 4, the kth joint essential maximal numerical range, spatial maximal numerical range and algebraic numerical range were introduced. It was shown that kth joint essential maximal numerical range is non-empty and convex. Also, it was shown that the kth joint algebraic maximal numerical range is the convex hull of the kth joint spatial maximal numerical range. This extends the corresponding result of Fong.

Halpern has defined a center valued essential spectrum, ΣI(A), and numerical range, Wʓ(A), for operators A in a von Neumann algebra ɸ. By restricting our attention to algebras ɸ which act on a separable Hilbert space, we can use a direct integral decomposition of ɸ to obtain simple characterizations of these qualities, and this in turn enables us to prove analogues of some classical results. since the essential spectrum is defined relative to a central ideal, we first show that, under the separability assumption, every ideal, modulo the center, is an ideal generated by finite projections. This leads to the following decomposition theorem: Theorem : Z = ʃΛ ⊕ c(λ)dµ ∈ ΣI(A) if and only if c(λ) ∈ σe(A(λ)) µ-a.e., where A = ʃΛ ⊕ A(λ)dµ and σe is a suitable spectrum in the algebra ɸ(λ). Using mainly measure-theoretic arguments, we obtain a similar decomposition result for the norm closure of the central numerical range: Theorem : Z = ʃΛ ⊕ c(λ)dµ ∈ Wʓ(A) if and only if c(λ) ∈ W(A(λ)) µ-a.e. By means of these theorems, questions about ΣI(A) and W (A) in ɸ can be reduced to the factors ɸ(λ). As examples, we show that spectral mapping holds for ΣI, namely f(ΣI(A)) = ΣI(f(A)), and that a generalization of the power inequality holds for Wʓ(A). Dropping the separability assumption, we show that central ideals can be defined in purely algebraic terms, and that the following perturbation result holds: Thereom : ΣI(A + X) = ΣI(A) for all A ∈ ɸ if and only if X ∈ I.

We analyse how the spectrum of the anisotropic Maxwell system with bounded conductivity σ on a Lipschitz domain Ω is approximated by domain truncation. First we prove a new non-convex enclosure for the spectrum of the Maxwell system, with weak assumptions on the geometry of Ω and none on the behaviour of the coefficients at infinity. We also establish a simple criterion for non-accumulation of eigenvalues at iR as well as resolvent estimates. For asymptotically constant coefficients, we describe the essential spectrum and show that spectral pollution may occur only in the essential numerical range We(L∞)⊂R of the quadratic pencil L∞(ω)=μ∞−1curl2−ω2ε∞, acting on divergence-free vector fields. Further, every isolated spectral point of the Maxwell system lying outside We(L∞) and outside the part of the essential spectrum on iR is approximated by spectral points of the Maxwell system on the truncated domains. Our analysis is based on two new abstract results on the (limiting) essential spectrum of polynomial pencils and triangular block operator matrices, which are of general interest. We believe our strategy of proof could be used to establish domain truncation spectral exactness for more general classes of non-self-adjoint differential operators and systems with non-constant coefficients.

On essential numerical ranges and essential spectra of Hamiltonian systems with one singular endpoint

This paper is devoted to the investigation of the stability of upper semi-Browder, lower semi-Browder and Browder linear relations which satisfy the stabilization property, under commuting Riesz operator perturbations. As applications, we infer the invariance of the corresponding Browder’s essential approximate point spectrum, Browder’s essential defect spectrum and Browder’s essential spectrum under commuting Riesz operator perturbations.

Weyl type theorems of [formula omitted] upper triangular operator matrices

A characterization of some subsets of Schechter's essential spectrum and application to singular transport equation

Browder and semi-browder operators

The concept of essential numerical range of an operator was dened and studied by Stamp i and Williams in 1972. Researchers generalised this idea of essential numerical range to a group of operators to the joint essential numerical range. In this paper, we consider the jointessential numerical range and show that the properties of the classical numerical range such as compactness also hold for the joint essential numerical range. Further, we show that the joint essential spectrum is contained in the joint essential numerical range by looking at the boundary of the joint essential spectrum.

Perturbation of semi-Browder operators and stability of Browder's essential defect and approximate point spectrum

A closed two-sided ideal ^f in a von Neumann algebra is defined to be a central ideal if ^ A%Pi is in ^ for every set {Pi} of orthogonal projections in the center %* of J ^ and every bounded subset {A*} of KJ'. Central ideals are characterized in terms of the existence of continuous fields and their form is completely determined. If ^ is a central ideal of Jzf and A e J ^ then Ao e %* is said to be in the essential central spectrum of A if Ao — A is not invertible in Sϊf modulo the smallest closed ideal containing ^ and ζ for every maximal ideal ζ of %*. It is shown that the essential central spectrum is a nonvoid, strongly closed subset of %? and that it satisfies many of the relations of the essential spectrum of operators on Hubert space. Let j y ~ be the space of all bounded ^-module homomorphisms of J ^ into -S. The essential central numerical range of i e with respect to ^ is defined to be Sέ^(A)={φ(A) φe ~, II Φ II ^ 1, 0(1) = IV, Φ(^) = (0)}. Here P ^ is the orthogonal complement of the largest central projection in *Jζ The essential central numerical range is shown to be a weakly closed, bounded, ^-convex subset of %£. It possesses many of the properties of the essential numerical range but in a form more suited to the fact that A is in Ssf rather than a bounded operator. It is shown that if Sf is properly infinite and ^ is the ideal of finite elements (resp. the strong radical) of J ^ then «-%S(A) is the intersection of JΓ with the weak (resp. uniform) closure of the convex hull of {UAU~ U unitary in

We analyze the notion of Bishop’s property ($\beta$) to obtain some new concepts. We describe some conditions in terms of these concepts for an operator to have its essential spectrum (spectrum) contained in the essential spectrum (spectrum) of every operator quasisimilar to it. A subfamily of such operators is proved to be dense in $L(\mathbf {H})$.

This paper focuses on exploring the relationship between the essential approximate point spectrum (and the essential defect spectrum) of a sequence of closedlinear operators (Tn)n2N acting on a Banach space X, and the corresponding spectraof a linear operator T on X. We examine this relationship under both generalizedconvergence and compact convergence conditions for the sequence (Tn)n2N convergingto T.

We consider the spectral stability problem for a family of standing pulse and wave front solutions to the one-dimensional version of the $M^5$-model formulated by Hillen [T. Hillen, $M^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585--616], to describe the mesenchymal cell motion inside tissue. The stability analysis requires the definition of spectrum, which is divided into two disjoint sets: the point spectrum and the essential spectrum. Under this partition the eigenvalue zero belongs to the essential spectrum and not to the point spectrum. By excluding the eigenvalue zero we can bring the spectral problem into an equivalent scalar quadratic eigenvalue problem. This leads, naturally, to deduce the existence of a negative eigenvalue which also turns out to belong to the essential spectrum. Beyond this result, the scalar formulation enables us to use the integrated equation technique to establish, via energy methods, that the point spectrum is empty. Our main result is that the family of standing waves is spectrally stable. To prove it, we go back to the original scalar problem and show that the rest of the essential spectrum is a subset of the open left-half complex plane.

There are a number of spectra studied in literature which do not fit into the axiomatic theory of Żelazko. This paper is an attempt to give an axiomatic theory for these spectra which, apart from the usual types of spectra, like one-sided, approximate point or essential spectra, include also the local spectra, the Browder spectrum and various versions of the Apostol spectrum (studied under various names, e.g. regular, semi-regular or essentially semi-regular). I. Basic properties of regularities The axiomatic theory of spectrum was introduced by W. Żelazko [21], see also S lodkowski and Żelazko [17]. He gave a classification of various types of spectra defined for commuting n-tuples of elements of a Banach algebra. The most important notion is that of subspectrum. All algebras in this paper are complex and unital. Denote by Inv(A) the set of all invertible elements in a Banach algebra A and by σ(a) = {λ ∈ C, a− λ / ∈ Inv(A)} the ordinary spectrum of an element a ∈ A. The spectral radius of a ∈ A will be denoted by r(a). Definition 1.1. Let A be a Banach algebra. A subspectrum σ in A is a mapping which assigns to every n-tuple (a1, . . . , an) of mutually commuting elements of A a non-empty compact subset σ(a1, . . . , an) ⊂ C such that (1) σ(a1, . . . , an) ⊂ σ(a1)× · · · × σ(an), (2) σ(p(a1, . . . , an)) = p(σ(a1, . . . , an)) for every commuting a1, . . . , an ∈ A and every polynomial mapping p = (p1, . . . , pm) : C → C. This notion has proved to be quite useful since it includes for example the left (right) spectrum, the left (right) approximate point spectrum, the Harte (= the union of the left and right) spectrum, the Taylor spectrum and various essential spectra. However, there are also many examples of spectrum, usually defined only for single elements of A, which are not covered by the axiomatic theory of Żelazko. The aim of this paper is to give an axiomatic description of such spectra. Instead of describing a spectrum, it is possible to describe equivalently the set of regular elements. Definition 1.2. Let A be a Banach algebra. A non-empty subset R of A is called a regularity if (1) if a ∈ A and n ∈ N then a ∈ A ⇔ a ∈ A, (2) if a, b, c, d are mutually commuting elements of A and ac + bd = 1A, then ab ∈ R ⇔ a ∈ R and b ∈ R.