The joint numerical range and the joint essential numerical range

Let T be a bounded linear operator on a complex separable Hilbert space . Let and be the Aluthge transform and *-Aluthge transform of T respectively. In this article, we consider the essential numerical range and maximal numerical range of T, and . We prove that the essential numerical range of is always contained in that of T and is the same as that of . Some related results for maximal numerical range are obtained.


 
 
 Our purpose is to study the relationship between the joint numerical range and joint essential numerical range. We give an example of an operator such that the set of all extreme points of the closure of its essential numerical range is not a subset of the set of all exti:eme points of its numerical range. We shall investigate the extreme points of a convex joint essential numerical range. 
 
 

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 introduce two new concepts for unbounded operators $T$ in a Hilbert space, the essential numerical range $W_{e5}(T)$ of type $5$ and the $C$-numerical range $W_C(T)$. Our first main result clarifies the relation of $W_{e5}(T)$ to the essential numerica

The numerical range in the quaternionic setting is, in general, a non-convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.

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.

A bounded linear operator acting on a Hilbert space is a generalized quadratic operator if it has an operator matrix of the form It reduces to a quadratic operator if d = 0. In this article, spectra, norms and various kinds of numerical ranges of generalized quadratic operators are determined. Some operator inequalities are also obtained. In particular, it is shown that for a given generalized quadratic operator, the rank-k numerical range, the essential numerical range and the q-numerical range are elliptical discs; the c-numerical range is a sum of elliptical discs. The Davis–Wielandt shell is the convex hull of a family of ellipsoids unless the underlying Hilbert space has dimension 2.

Real higher rank numerical ranges and ellipsoids

ABSTRACTLet H be a complex Hilbert space and let be an n-tuple of self-adjoint operators on H. The joint numerical range of is the setBy a result of Li and Poon, the closure of , denoted by , is star-shaped with each point in the joint essential numerical range of a star centre. In this note, we give a few observations about the boundary of and basing on their results.

Let be a complex infinite-dimensional Hilbert space, let be a bounded linear operator on , and let . The k-numerical range of T, denoted by Wk (T), is defined to be the set of points obtained by letting (e1 ,…,ek ) vary over the set of all orthonormal k-tuples in Let We (T) denote the essential numerical range of T. Let and let {βij} i=1 be a sequence of non-negative scalars such that the series converges. In this paper we show that if, and only if,.

Let $D$ be a bounded convex domain in $\mathbb{C}$ with a regular analytic boundary. Suppose that the numerical range $W(A)$ of a bounded linear operator $A$ is contained in $\overline{D}$. If $\overline{W(A)}$ intersects the boundary $\partial D$ at infinitely many points while the essential numerical range $W_\text{ess}(A)$ does not intersect $\partial D$, then $W(A) = \overline{D}$. This generalizes some infinite dimensional analogues of a result of Anderson.

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In this paper we discuss the relationship between the numerical range of an extensive class of unbounded operator functions and the joint numerical range of the operator coefficients. Furthermore, we derive methods on how to find estimates of the joint numerical range. Those estimates are used to obtain explicitly computable enclosures of the numerical range of the operator function and resolvent estimates. The enclosure and upper estimate of the norm of the resolvent are optimal given the estimate of the joint numerical range.

Let W(A) and W(e)(A) be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A(1),.., A(m)) acting on an infinite-dimensional Hilbert space. It is shown that W(e)(A) is always convex and admits many equivalent formulations. In particular, for any fixed i is an element of {1,..., m}, W(e) (A) can be obtained as the intersection of all sets of the form cl(W(A(1),..., A(i+1), A(i) + F, A(i+1),..., A(m))), where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in W(e)(A) as star centers. Although cl(W(A)) is usually not convex, an analog of the separation theorem is obtained, namely, for any element d is not an element of cl(W(A)), there is a linear functional f such that f (d) > sup{f (a) : a is an element of cl(W((A) over tilde))}, where (A) over tilde is obtained from A by perturbing one of the components A(i) by a finite rank self-adjoint operator. Other results on W(A) and W(e)(A) extending those on a single operator axe obtained.

This paper focuses on the properties of the essential maximal numerical range of Aluthgetransform T. For instance, among other results, we show that the essential maximal numerical rangeof Aluthge transform is nonempty and convex. Further, we prove that the essential maximal numericalrange of Aluthge transform e T is contained in the essential maximal numerical range of T. This studyis therefore an extention of the research on Aluthge transform which was begun by Aluthge in hisstudy of p−hyponormal operators.