scholarly journals Polynomials in a hermitian element

1988 ◽  
Vol 30 (2) ◽  
pp. 171-176 ◽  
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Dual Space ◽  
Numerical Range ◽  
Numerical Radius ◽  
Hermitian Element ◽  
Unital Banach Algebra

For an element a of a unital Banach algebra A with dual space A′, we define the numerical range V(a) = {f(a):f ∊ A′, ∥f∥ = f(1) = 1}, and the numerical radius v(a) = sup{⃒z⃒:z ∊ V(a)}. An element a is said to be Hermitian if V(a) ⊆ ℝ ,equivalently ∥exp (ita)∥ = 1(t ∊ ℝ). Under the condition V(h) ⊆ [-1, 1], any polynomial in h attains its greatest norm in the algebra Ea[-1,1], generated by an element h with V(h) = [-1, 1].

scholarly journals The power inequality on normed spaces

1971 ◽  
Vol 17 (3) ◽  
pp. 237-240 ◽  
Author(s):  
Michael J. Crabb
Keyword(s):  
Linear Operator ◽  
Normed Space ◽  
Bounded Linear Operator ◽  
Dual Space ◽  
Numerical Range ◽  
Normed Spaces ◽  
Numerical Radius ◽  
Bounded Linear ◽  
Power Inequality ◽  
Unital Banach Algebra

Let X be a complex normed space, with dual space X′. Let T be a bounded linear operator on X. The numerical range V(T) of T is defined as {f(Tx): x ∈ X, f ∈ X′, ‖ x ‖ = ‖ f ‖ = f(x) = 1}, and the numerical radius v(T) of T is defined as sup {|z|: z ∈ V(T)}. For a unital Banach algebra A, the numerical range V(a) of a ∈ A is defined as V(Ta), where Ta is the operator on A defined by Tab = ab. It is shown in (2, Chapter 1.2, Lemma 2) that V(a) = {f(a): f ∈ D(1)}, where D(1) = {f ∈ A′: ‖f‖ = f(1) = 1}.

On Banach algebra elements of thin numerical range

1979 ◽  
Vol 86 (1) ◽  
pp. 71-83 ◽  
Author(s):  
R. R. Smith
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Natural Generalization ◽  
The Real ◽  
C Algebra ◽  
Image Position ◽  
Unital Banach Algebra ◽  
Real Subspace

Among the elements of a complex unital Banach algebra the real subspace of hermitian elements deserves special attention. This forms the natural generalization of the set of self-adjoint elements in a C*-algebra and exhibits many of the same properties. Two equivalent definitions may be given: if W(h) ⊂ , where W(h) denotes the numerical range of h (7), or if ║eiλh║ = 1 for all λ ∈ . In this paper some related subsets are introduced and studied. For δ ≥ 0, an element is said to be a member of if the conditionis satisfied. These may be termed the elements of thin numerical range if δ is small.

scholarly journals Numerical ranges of powers of hermitian elements

1986 ◽  
Vol 28 (1) ◽  
pp. 37-45 ◽  
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Previous Knowledge ◽  
Line Segments ◽  
Straight Line ◽  
Image Position ◽  
Unital Banach Algebra

An element k of a unital Banach algebra A is said to be Hermitian if its numerical rangeis contained in ℝ; equivalently, ∥eitk∥ = 1(t ∈ ℝ)—see Bonsall and Duncan [3] and [4]. Here we find the largest possible extent of V(kn), n ∈ ℕ, given V(k) ⊆ [−1, 1], and so ∥k∥ ≤ 1: previous knowledge is in Bollobás [2] and Crabb, Duncan and McGregor [7]. The largest possible sets all occur in a single example. Surprisingly, they all have straight line segments in their boundaries. The example is in [2] and [7], but here we give A. Browder's construction from [5], partly published in [6]. We are grateful to him for a copy of [5], and for discussions which led to the present work. We are also grateful to J. Duncan for useful discussions.

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 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].

The power inequality on Banach spaces

1971 ◽  
Vol 69 (3) ◽  
pp. 411-415 ◽  
Author(s):  
Béla Bollobás
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Normed Space ◽  
Bounded Linear Operator ◽  
Dual Space ◽  
Numerical Range ◽  
Numerical Radius ◽  
Bounded Linear ◽  
Power Inequality ◽  
Image Position

Let X be a complex normed space with dual space X′ and let T be a bounded linear operator on X. The numerical range of T is defined asand the numerical radius is v(T) = sup {|ν: νε V(T)}. Most known results and problems concerning numerical range can be found in the notes by Bonsall and Duncan (5).

The numerical range in the second dual of a Banach algebra

1981 ◽  
Vol 89 (2) ◽  
pp. 301-307
Author(s):  
R. R. Smith
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Numerical Range ◽  
Special Property ◽  
Empty Interior ◽  
Banach Theorem ◽  
Second Dual ◽  
Algebra Theory ◽  
Unital Banach Algebra ◽  
Selection Of

An elementary consequence of the Hahn-Banach theorem is that every Banach space Y is ω*-dense in its second dual Y**, so that every element y ∈ Y** is the w*-limit of a net {ya}α ∈ Λ from Y. There is, of course, a great deal of choice in the selection of such a net, and so one may impose extra conditions on the net related to some special property of the limit point, and then ask for existence. The object of this paper is to present such a result in the context of a unital Banach algebra and its second dual , and then to give two applications to Banach algebra theory. The theorem to be proved is this: if the numerical range W(a) of an element in has non-empty interior then a is the ω*-limit of a net {aa}α ∈ Λ from whose numerical ranges are contained in W(a), while if W(a) has empty interior then the numerical ranges W(aα) are contained in a shrinking set of neighbourhoods of W(a).

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

Sign in / Sign up

Export Citation Format

Share Document