Numerical ranges and dilations

Denote by W(A) the numerical range of a bounded linear operator A. For two operators A and B (which may act on different Hilbert spaces), we study the relation between the inclusion relation W(A)⊆W(B) and the condition that A can be dilated to an operator of the form B⊗I. We also investigate the possibilities of dilating an operator A to operators with simple structure under the assumption that W(A) is included in a special region.     

Numerical ranges of weighted composition operators

The operator that takes the function f   to ψf°φ$\psi f°\phi$ is a weighted composition operator. We study numerical ranges of some classes of weighted composition operators on H²$H^2$, the Hardy–Hilbert space of the unit disc. We consider the case where φ is a rotation of the unit disc and identify a class of convexoid operators. In the case of isometric weighted composition operators we give a complete classification of their numerical ranges. We also consider the inclusion of zero in the interior of the numerical range.     

Numerical ranges of the powers of an operator

The numerical range W(A) of a bounded linear operator A on a Hilbert space is the collection of complex numbers of the form (Av,v) with v ranging over the unit vectors in the Hilbert space. In terms of the location of W(A), inclusion regions are obtained for W(Ak) for positive integers k, and also for negative integers k if A−1 exists. Related inequalities on the numerical radius and the Crawford number are deduced.     

On numerical ranges and roots

Existence of the fractional powers is established in Banach algebra setting, in terms of the numerical ranges of elements involved. The behavior of the spectra and (for Hermitian ∗-algebras satisfying some additional hypotheses) the ∗-numerical range under taking these powers also is investigated.     

Numerical ranges of weighted shifts

Let A be a unilateral (resp., bilateral) weighted shift with weights wn, n?0 (resp., −∞<n<∞). Eckstein and Rácz showed before that A has its numerical range W(A) contained in the closed unit disc if and only if there is a sequence (resp., ) in [−1,1] such that ²|wn|=(1−an)(1+an+1) for all n. In terms of such an?s, we obtain a necessary and sufficient condition for W(A) to be open. If the wn?s are periodic, we show that the an?s can also be chosen to be periodic. As a result, we give an alternative proof for the openness of W(A) for an A with periodic weights, which was first proven by Stout. More generally, a conjecture of his on the openness of W(A) for A with split periodic weights is also confirmed.     

Numerical ranges in physics

Some examples of the incidence of Numerical Ranges in Physics are discussed. The Thermodynamical Inequality is deduced using the theory of Numerical Ranges. The Peierls-Bogoliubov inequality is obtained from the Thermodynamical Inequality. The formalism of Thermo Field Dynamics is summarized in the same framework.     

On numerical ranges of the compressions of normal matrices

For an n × n normal matrix A, whose numerical range NR[A] is a k-polygon (k ? n), an n × (k − 1) isometry matrix P is constructed by a unit vector υ∈Cⁿ, and NR[P∗AP] is inscribed to NR[A]. In this paper, using the notations of NR[P∗AP] and some properties from projective geometry, an n × n diagonal matrix B and an n × (k − 2) isometry matrix Q are proposed such that NR[P∗AP] and NR[Q∗BQ] have as common support lines the edges of the k-polygon and share the same boundary points with the polygon. It is proved that the boundary of NR[P∗AP] is a differentiable curve and the boundary of the numerical range of a 3 × 3 matrix P∗AP is an ellipse, when the polygon is a quadrilateral.     

Anderson's theorem for compact operators

It is shown that if is a compact operator on a Hilbert space with its numerical range contained in the closed unit disc and with intersecting the unit circle at infinitely many points, then is equal to . This is an infinite-dimensional analogue of a result of Anderson for finite matrices.

     

Some results on numerical ranges and factorizations of matrix polynomials

Some new factorization theorems for monic matrix polynomials are obtained. These theorems are based on the numerical range having the number of connected components equal to the degree of the polynomial. For second degree polynomials, sufficient conditions are given for the numerical range to have two connected components.     

Volterra算子及其伴随算子线性组合的数值域

经典Volterra算子$V$及其伴随算子$V^*$在复空间$L^2[0,1]$中起着关键作用. 关于$V$和$V^*$的线性组合的性质, 我们给出了确保$z_1V+z_2V^*(z_1,z_2\in\mathbb{C})$满足增生性质的等价条件. 本文描述了$(u+iv)I+mV+nV^*(u,v,m,n\in\mathbb{R},m+n\geq0)$数值域的精确表示.     

Remarks on numerical ranges of operators in spaces with an indefinite metric

The numerical range of an operator on an indefinite inner product space (possibly infinite dimensional) is studied. In particular, operators having bounded numerical ranges are characterized, and the angle points of the numerical range and their connections with eigenvalues are described.

     

Congruence numerical ranges and their radii

The earliest congruence numerical range was introduced by Thompson in 1980. Since then, congruence numerical ranges and their radii have been studied by some authors. The purpose of this article is to discuss several aspects of this subject. We survey some known results and propose some problems for furthur study. A few new results are also presented.     

Congruence numerical ranges and their radii

The earliest congruence numerical range was introduced by Thompson in 1980. Since then, congruence numerical ranges and their radii have been studied by some authors. The purpose of this article is to discuss several aspects of this subject. We survey some known results and propose some problems for furthur study. A few new results are also presented.     

Numerical index and renorming

We study the numerical index of a Banach space from the isomorphic point of view, that is, we investigate the values of the numerical index which can be obtained by renorming the space. The set of these values is always an interval which contains in the real case and in the complex case. Moreover, for ``most' Banach spaces the least upper bound of this interval is as large as possible, namely .

     

The numerical range of a nilpotent operator on a Hilbert space

We prove that the numerical range of an arbitrary nilpotent operator on a complex Hilbert space is a circle (open or closed) with center at and radius not exceeding where is the power of nilpotency of

     

Norms of certain Jordan elementary operators

Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. For A,B∈B(H), the Jordan elementary operator UA,B is defined by UA,B(X)=AXB+BXA, ∀X∈B(H). In this short note, we discuss the norm of UA,B. We show that if dimH=2 and ‖UA,B‖=‖A‖‖B‖, then either AB^∗ or B^∗A is 0. We give some examples of Jordan elementary operators UA,B such that ‖UA,B‖=‖A‖‖B‖ but AB^∗≠0 and B^∗A≠0, which answer negatively a question posed by M. Boumazgour in [M. Boumazgour, Norm inequalities for sums of two basic elementary operators, J. Math. Anal. Appl. 342 (2008) 386-393].     

The numerical range of a tridiagonal operator

Let r be a real number and A a tridiagonal operator defined by Aej=ej−1+rjej+1, j=1,2,…, where {e₁,e₂,…} is the standard orthonormal basis for ?²(N). Such tridiagonal operators arise in Rogers-Ramanujan identities. In this paper, we study the numerical ranges of these tridiagonal operators and finite-dimensional tridiagonal matrices. In particular, when r=−1, the numerical range of the finite-dimensional tridiagonal matrix is the convex hull of two explicit ellipses. Applying the result, we obtain that the numerical range of the tridiagonal operator is the square

     

Hamilton算子的一类n次数值域的对称性

以Hamilton算子的数值域为基础,研究了一类算子的二次数值域关于实轴,虚轴的对称性.此外从α-J-自伴算子的n次数值域关于过原点直线对称出发,得到了有界Hamilton算子的一类n次数值域关于虚轴的对称性.     

On the Origin and Numerical Range of Bounded Operators

Numerical range has an important applications on spectrum distribution of operators. In this paper, we devoted to characterizing operators whose numerical range contains the origin. Some necessary and sufficient conditions are given by operator decomposition technique and constructive methods. Furthermore, the closeness of the numerical range of a given operator is also investigated.