scholarly journals Some extremal algebras for hermitians

2001 ◽  
Vol 43 (1) ◽  
pp. 29-38
Author(s):  
M. J. Crabb ◽  
J. Duncan ◽  
C. M. McGregor
Keyword(s):  
Numerical Range ◽  
Banach Algebras ◽  
The Third

We study three extremal Banach algebras: (a) generated by two hermitian unitaries; (b) generated by an element of norm 1 all of whose odd positive powers are hermitian; (c) generated by an element of norm 1 all of whose even positive powers are hermitian. In all three cases the numerical range is found for various elements. The second algebra is shown to be isometrically isomorphic to a subalgebra of the first. The third algebra is identified with a space of functions.

scholarly journals Functional Identities and Zero Lie Product Determined Banach Algebras

2020 ◽  
Vol 71 (2) ◽  
pp. 649-665
Author(s):  
Matej Brešar
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
The Third ◽  
Linear Maps ◽  
Functional Identities

Abstract Three problems connecting functional identities to the recently introduced notion of a zero Lie product determined Banach algebra are discussed. The first one concerns commuting linear maps, the second one concerns derivations that preserve commutativity and the third one concerns bijective commutativity preserving linear maps.

scholarly journals A characterization of Banach-star-algebras by numerical range

1971 ◽  
Vol 4 (2) ◽  
pp. 193-200 ◽  
Author(s):  
Brailey Sims
Keyword(s):  
Elementary Proof ◽  
Numerical Range ◽  
Banach Algebras

It is known that in a B*-algebra every self-adjoint element is hermitian. We give an elementary proof that this condition characterizes B*-algetras among Banach*-algebras.

scholarly journals A non-smooth extension of Frechet differentiability of the norm with applications to numerical ranges

1986 ◽  
Vol 28 (2) ◽  
pp. 121-137 ◽  
Author(s):  
C. Aparicio ◽  
F. Ocaña ◽  
R. Payá ◽  
A. Rodríguez
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Banach Algebras ◽  
Continuous States ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Smooth Extension ◽  
Unital Banach Algebra

The following result in the theory of numerical ranges in Banach algebras is well known (see [3, Theorem 12.2]). The numerical range of an element F in the bidual of a unital Banach algebra A is the closure of the set of values at F of the w*-continuous states of . As a consequence of the results in this paper the following

scholarly journals Interpolation and inequalities for functions of exponential type: the Arens irregularity of an extremal algebra

1993 ◽  
Vol 35 (3) ◽  
pp. 325-326
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Exponential Type ◽  
Line Segment ◽  
Convex Set ◽  
Numerical Range ◽  
Compact Convex ◽  
Banach Algebras ◽  
Functions Of Exponential Type ◽  
Compact Convex Set ◽  
Unital Banach Algebra ◽  
Extremal Algebra

For any compact convex set K ⊂ ℂ there is a unital Banach algebra Ea(K) generated by an element h in which every polynomial in h attains its maximum norm over all Banach algebras subject to the numerical range V(h) being contained in K, [1]. In the case of K a line segment in ℝ, we show here that Ea(K) does not have Arens regular multiplication. We also show that ideas about Ea(K) give simple proofs of, and extend, two inequalities of C. Frappier [4].

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.

scholarly journals Imbedding elements whose numerical range has a vertex at zero in holomorphic semigroups

1985 ◽  
Vol 28 (1) ◽  
pp. 91-95
Author(s):  
J. Martinez-Moreno ◽  
A. Rodriguez-Palacios
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Compact Convex ◽  
Banach Algebras ◽  
Point Of View ◽  
Holomorphic Semigroup ◽  
Compact Convex Set ◽  
Basic Ideas ◽  
Unital Banach Algebra ◽  
First Time

If a is an element of a complex unital Banach algebra whose numerical range is confined to a closed angular region with vertex at zero and angle strictly less than π, we imbed a in a holomorphic semigroup with parameter in the open right half plane.There has been recently a great development in the theory of semigroups in Banach algebras (see [6]), with attention focused on the relation between the structure of a given Banach algebra and the existence of continuous or holomorphic non-trivial semigroups with certain properties with range in this algebra. The interest of this paper arises from the fact that we relate in it, we think for the first time, this new point of view in the theory of Banach algebras with the already classic one of numerical ranges [2,3]. The proofs of our results use, in addition to some basic ideas from numerical ranges in Banach algebras, the concept of extremal algebra Ea(K) of a compact convex set K in ℂ due to Bollobas [1] and concretely the realization of Ea(K) achieved by Crabb, Duncan and McGregor [4].

scholarly journals Eigenvalue inequalities for positive block matrices with the inradius of the numerical range

2022 ◽  
Author(s):  
Jean-Christophe Bourin ◽  
Eun-Young Lee
Keyword(s):  
Numerical Range ◽  
Norm Inequality ◽  
Operator Norm ◽  
Diagonal Block ◽  
Positive Matrix ◽  
Full Matrix ◽  
Block Matrices ◽  
The Third ◽  
Four Blocks ◽  
Second Eigenvalue

We prove the operator norm inequality, for a positive matrix partitioned into four blocks in [Formula: see text], [Formula: see text] where [Formula: see text] is the diameter of the largest possible disc in the numerical range of [Formula: see text]. This shows that the inradius [Formula: see text] satisfies [Formula: see text] Several eigenvalue inequalities are derived. In particular, if [Formula: see text] is a normal matrix whose spectrum lies in a disc of radius [Formula: see text], the third eigenvalue of the full matrix is bounded by the second eigenvalue of the sum of the diagonal block, [Formula: see text] We think that [Formula: see text] is optimal and we propose a conjecture related to a norm inequality of Hayashi.

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