scholarly journals The numerical range of some periodic tridiagonal operators is the convex hull of the numerical ranges of two finite matrices

2021 ◽  
pp. 1-20
Author(s):  
Benjamín A. Itzá-Ortiz ◽  
Rubén A. Martínez-Avendaño ◽  
Hiroshi Nakazato
Keyword(s):  
Convex Hull ◽  
Numerical Range

scholarly journals The boundary of the numerical range

1981 ◽  
Vol 22 (1) ◽  
pp. 69-72 ◽  
Author(s):  
G. de Barra
Keyword(s):  
Hilbert Space ◽  
Convex Hull ◽  
Similar Condition ◽  
Point Spectrum ◽  
Numerical Range ◽  
Normal Operator ◽  
Density Condition ◽  
Compact Normal Operator

In [1] it was shown that for a compact normal operator on a Hilbert space the numerical range was the convex hull of the point spectrum. Here it is shown that the same holds for a semi-normal operator whose point spectrum satisfies a density condition (Theorem 1). In Theorem 2 a similar condition is shown to imply that the numerical range of a semi-normal operator is closed. Some examples are given to indicate that the condition in Theorem 1 cannot be relaxed too much.

The numerical range of functions and best approximation

1974 ◽  
Vol 76 (1) ◽  
pp. 133-141 ◽  
Author(s):  
Lawrence A. Harris
Keyword(s):  
Convex Hull ◽  
Linear Space ◽  
Best Approximation ◽  
Normed Linear Space ◽  
Numerical Range ◽  
Closed Convex Hull ◽  
Compact Spaces ◽  
Range Of Functions ◽  
Function Mapping

In this note, we state general conditions which imply that the numerical range of a function mapping a set S into a normed linear space Y is the closed convex hull of the spatial numerical range of the function. This conclusion is of interest since, for example, it is equivalent to an extension to non-compact spaces of Kolmogoroff's characterization of functions of best approximation.

scholarly journals Spatial numerical ranges of elements of Banach algebras

1989 ◽  
Vol 12 (4) ◽  
pp. 633-640 ◽  
Author(s):  
A. K. Gaur ◽  
T. Husain
Keyword(s):  
Convex Hull ◽  
Banach Algebra ◽  
Numerical Range ◽  
Banach Algebras ◽  
Hermitian Element ◽  
Regular Norm ◽  
The Relationship

In this paper, the notion of spatial numerical range of elements of Banach algebras without identity is studied. Specifically, the relationship between spatial numerical ranges, numerical ranges and spectra is investigated. Among other results, it is shown that the closure of the spatial numerical range of an element of a Banach algebra without Identity but wlth regular norm is exactly its numerical range as an element of the unitized algebra. Futhermore, the closure of the spatial numerical range of a hermitian element coincides with the convex hull of its spectrum. In particular, spatial numerical ranges of the elements of the Banach algebraC0(X)are described.

C*-Convexity and the Numerical Range

2000 ◽  
Vol 43 (2) ◽  
pp. 193-207 ◽  
Author(s):  
Bojan Magajna
Keyword(s):  
Convex Hull ◽  
Extreme Point ◽  
Von Neumann Algebra ◽  
Numerical Range ◽  
Von Neumann ◽  
Norm Closure

AbstractIf A is a prime C*-algebra, a ∈ A and λ is in the numerical range W(a) of a, then for each ε > 0 there exists an element h ∈ A such that . If λ is an extreme point of W(a), the same conclusion holds without the assumption that A is prime. Given any element a in a von Neumann algebra (or in a general C*-algebra) A, all normal elements in the weak* closure (the norm closure, respectively) of the C*-convex hull of a are characterized.

scholarly journals Joint spectra and joint numerical ranges for pairwise commuting operators in Banach spaces

1988 ◽  
Vol 30 (2) ◽  
pp. 145-153 ◽  
Author(s):  
Volker Wrobel
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Numerical Range ◽  
Linear Operators ◽  
Continuous Linear ◽  
Joint Spectrum ◽  
Commuting Operators ◽  
Continuous Linear Operators

In a recent paper M. Cho [5] asked whether Taylor's joint spectrum σ(a1, …, an; X) of a commuting n-tuple (a1,…, an) of continuous linear operators in a Banach space X is contained in the closure V(a1, …, an; X)- of the joint spatial numerical range of (a1, …, an). Among other things we prove that even the convex hull of the classical joint spectrum Sp(a1, …, an; 〈a1, …, an〉), considered in the Banach algebra 〈a1, …, an〉, generated by a1, …, an, is contained in V(a1, …, an; X)-.

Some Convexity Features Associated with Unitary Orbits

2003 ◽  
Vol 55 (1) ◽  
pp. 91-111 ◽  
Author(s):  
Man-Duen Choi ◽  
Chi-Kwong Li ◽  
Yiu-Tung Poon
Keyword(s):  
Convex Hull ◽  
Linear Space ◽  
Spectral Properties ◽  
Numerical Range ◽  
Symmetric Matrices ◽  
Hermitian Matrices ◽  
Linear Transformations ◽  
Unitary Similarity ◽  
Similarity Orbit ◽  
Real Linear

AbstractLet be the real linear space of n × n complex Hermitian matrices. The unitary (similarity) orbit of C ∈ is the collection of all matrices unitarily similar to C. We characterize those C ∈ such that every matrix in the convex hull of can be written as the average of two matrices in . The result is used to study spectral properties of submatrices of matrices in , the convexity of images of under linear transformations, and some related questions concerning the joint C-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

Essentially Convexoid Operators

1981 ◽  
Vol 33 (2) ◽  
pp. 257-274
Author(s):  
Takayuki Furuta
Keyword(s):  
Hilbert Space ◽  
Convex Hull ◽  
Numerical Range ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Quotient Mapping ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Essentially Normal

Let H be a separable complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. Let π be the quotient mapping from B(H) onto the Calkin algebra B(H)/K(H), where K(H) denotes all compact operators on B(H). An operator T ∈ B(H) is said to be convexoid[14] if the closure of its numerical range W(T) coincides with the convex hull co σ(T) of its spectrum σ(T). T ∈ B(H) is said to be essentially normal, essentially G1, or essentially convexoid if π(T) is normal, G1 or convexoid in B(H)/K(H) respectively.

scholarly journals Numerical Range of the Derivation of an Induced Operator

2007 ◽  
Vol 82 (3) ◽  
pp. 325-344
Author(s):  
Randall R. Holmes ◽  
Chi-Kwong Li ◽  
Tin-Yau Tam
Keyword(s):  
Convex Hull ◽  
Symmetric Group ◽  
Irreducible Character ◽  
Product Space ◽  
Numerical Range ◽  
Inner Product Space ◽  
Inner Product ◽  
Complex Numbers ◽  
Derivation Operator ◽  
Induced Operator

AbstractLet V be an n–dimensional inner product space over , let H be a subgroup of the symmetric group on {l,…, m}, and let x: H → be an irreducible character. Denote by (H) the symmetry class of tensors over V associated with H and x. Let K (T) ∈ End((H)) be the operator induced by T ∈ End(V), and let DK(T) be the derivation operator of T. The decomposable numerical range W*(DK(T)) of DK(T) is a subset of the classical numerical range W(DK(T)) of DK(T). It is shown that there is a closed star-shaped subset of complex numbers such that⊆ W*(DK(T)) ⊆ W(DK(T)) = con where conv denotes the convex hull of . In many cases, the set is convex, and thus the set inclusions are actually equalities. Some consequences of the results and related topics are discussed.

scholarly journals A class of tridiagonal operators associated to some subshifts

2016 ◽  
Vol 14 (1) ◽  
pp. 352-360 ◽  
Author(s):  
Christian Hernández-Becerra ◽  
Benjamín A. Itzá-Ortiz
Keyword(s):  
Convex Hull ◽  
Numerical Range ◽  
The Other ◽  
Limit Operator ◽  
The Given

AbstractWe consider a class of tridiagonal operators induced by not necessary pseudoergodic biinfinite sequences. Using only elementary techniques we prove that the numerical range of such operators is contained in the convex hull of the union of the numerical ranges of the operators corresponding to the constant biinfinite sequences; whilst the other inclusion is shown to hold when the constant sequences belong to the subshift generated by the given biinfinite sequence. Applying recent results by S. N. Chandler-Wilde et al. and R. Hagger, which rely on limit operator techniques, we are able to provide more general results although the closure of the numerical range needs to be taken.

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