On the Maximal Numerical Range of Elementary Operators (original) (raw)

On the Numerical Range and Norm of Elementary Operators

Linear and Multilinear Algebra, 2004

W 0 ðR A, B J j Þ where VðÁÞ is the joint spatial numerical range, W 0 ðÁÞ is the algebraic numerical range and J is a norm ideal of BðEÞ: We shall show that this inclusion becomes an equality when R A, B is taken to be a derivation. Also, we deduce that wðU A, B J j Þ ! 2ð ffiffi ffi 2 p À 1ÞwðAÞwðBÞ, for A, B 2 BðEÞ and J is a norm ideal of BðEÞ, where wðÁÞ is the numerical radius. On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimate bound kU A, B jJk ! 2ð ffiffi ffi 2 p À 1ÞkAkkBk holds, if one of the following two conditions is satisfied: (i) J is a standard operator algebra of BðEÞ and A, B 2 J: (ii) J is a norm ideal of BðEÞ and A, B 2 BðEÞ:

The numerical range of elementary operators II

Linear Algebra and its Applications, 2001

For A, B ∈ L(H) (the algebra of all bounded linear operators on the Hilbert space H), it is proved that: (i) the generalized derivation δ A,B is convexoid if and only if A and B are convexoid; (ii) the operators δ A,B and δ A,B |C p (where p 1) have the same numerical range and are equal to W 0 (A) − W 0 (B) (where C p is the Banach space of the p-Schatten class operators on H).

A note on the maximal numerical range

Operators and Matrices, 2019

We show that the maximal numerical range of an operator has a non-empty intersection with the boundary of its numerical range if and only if the operator is normaloid. A description of this intersection is also given. First, let us set some notation and terminology. For a subset X of the complex plane C, by cl X, ∂X, and conv X we will denote the closure, boundary, and the convex hull of X, respectively. By an "operator" we throughout the paper understand a bounded linear operator acting on a Hilbert space H. The numerical range of such an operator A is defined by the formula W (A) = { Ax, x : x ∈ H, x = 1}, where .,. and. stand, respectively, for the scalar product on H and the norm associated with it. Introduced a century ago in the works by Toeplitz [8] and Hausdorrf [6] (and thus also known as the Toeplitz-Hausdorff set), it since has been a subject of intensive research. We mention here only [4] as a standard source of references, and note the following basic properties: Due to the Cauchy-Schwarz inequality, the set W (A) is bounded. Namely, w(A) := sup{|z| : z ∈ W (A)} ≤ A ; (1) w(A) is called the numerical radius of A.

On Generalized Numerical Ranges of Operators on an Indefinite Inner Product Space

Linear and Multilinear Algebra, 2004

In this paper, numerical ranges associated to operators on an indefinite inner product space are investigated. Boundary generating curves, corners, shapes and computer generations of these sets are studied. In particular, the Murnaghan-Kippenhahn theorem for the classical numerical range is generalized.

On operators preserving the numerical range

Linear Algebra and its Applications, 1990

Let F be a surjective linear mapping between the algebras L(H) and L(K) of all bounded operators on nontrivial complex Hilbert spaces H and K respectively. For any positive integer k let W,(A) denote the kth numerical range of an operator A on H. If k is strictly less than one-half the dimension of H and W,(F(A)) = Wk. A) for ah A from L(H), then there is a unitary mapping U: H + K such that either F(A) = UAu* or F(A) = (UAU*)' for every A E L(H), where the transposition is taken in any basis of K, fixed in advance. This generalizes the result of S. Pierce and W. Watkins on finite-dimensional spaces. The case of k greater than or equal to one-half of the dimension of H is also treated using our method. Our proofs depend on a characterization of those linear operators preserving projections of rank one, which is of independent interest.

The joint essential numerical range of operators: convexity and related results

2009

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, respectively. In this paper, it is shown that W e (A) is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ {1, . . . , m}, W e (A) can be obtained as the intersection of all sets of the form

On the range of elementary operators

Integral Equations and Operator Theory, 2005

Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Let A = (A1, A2, .., An) and B = (B1, B2, .., Bn) be n-tuples in B(H), we define the elementary operator EA,B : B(H) → B(H) by EA,B(X) = È n i=1 AiXBi.

An Application Of Maximal Numerical Range On Norm Of Basic Elementary Operator In Tensor Product

Journal of Progressive Research in Mathematics, 2022

This study 1.0 Introduction 1.1 Tensor products of Hilbert spaces Definition 1.1.1. Tensor product. (Muiruri et al, 2019) If = { 1, 2 …} and = { 1 , 2 … } are complex Hilbert spaces. Define their inner products < 1 2 > and < 2 2 > respectively. A tensor product of and is a Hilbert space ⊗ where ⊗: × → ⊗ ,⊗ , → ⊗ is a bilinear mapping: i).