The Spectral Radius of a Hermitian Element of a Banach Algebra

1970 ◽  
Vol 2 (2) ◽  
pp. 178-180 ◽  
Author(s):  
F. F. Bonsall ◽  
M. J. Crabb
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Hermitian Element

scholarly journals Banach algebra elements whose spectra intersect R+

1974 ◽  
Vol 19 (1) ◽  
pp. 59-69 ◽  
Author(s):  
F. F. Bonsall ◽  
A. C. Thompson
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Real Numbers ◽  
Complex Banach Algebra ◽  
Image Position ◽  
Invertible Elements

Let A denote a complex Banach algebra with unit, Inv(A) the set of invertible elements of A, Sp(a) and r(a) the spectrum and spectral radius respectively of an element a of A. Let Γ denote the set of elements of A whose spectra contain non-negative real numbers, i.e.

scholarly journals Spectral radius formulae

1979 ◽  
Vol 22 (3) ◽  
pp. 271-275 ◽  
Author(s):  
G. J. Murphy ◽  
T. T. West
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Quotient Algebra ◽  
Complex Banach Algebra ◽  
Image Position

If A is a complex Banach algebra (not necessarily unital) and x∈A, σ(x) will denote the spectrum and spectral radius of x in A. If I is a closed two-sided ideal in A let x + I denote the coset in the quotient algebra A/I containing x. Then

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].

Spectrally bounded traces on C*-algebras

2003 ◽  
Vol 68 (1) ◽  
pp. 169-173 ◽  
Author(s):  
Martin Mathieu
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Linear Mapping ◽  
C Algebras ◽  
Spectrally Bounded ◽  
Bounded Trace

A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is called spectrally bounded if there is a constant M ≥ 0 such that r(T x) ≤ Mr(x) for all x ∈ E, where r (·) denotes the spectral radius. We establish the equivalence of the following properties of a unital linear mapping T from a unital C* -algebra A into its centre:(a) T is spectrally bounded;(b) T is a spectrally bounded trace;(c) T is a bounded trace.

scholarly journals Decomposition algebras of Riesz operators

1980 ◽  
Vol 21 (1) ◽  
pp. 75-79 ◽  
Author(s):  
G. J. Murphy ◽  
T. T. West
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Spectral Radius ◽  
Compact Operators ◽  
Linear Operators ◽  
Closed Ideal ◽  
Canonical Mapping ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Riesz Operators

Let H be a Hilbert space and let B denote the Banach algebra of all bounded linear operators on H with K denoting the closed ideal of compact operators in B. If T ∈ B, σ(T) and r(T) will denote the spectrum and spectral radius of T, respectively, and π the canonical mapping of B onto the Calkin algebra B/K.

scholarly journals Formulas for the joint spectral radius of noncommuting Banach algebra elements

1995 ◽  
Vol 123 (9) ◽  
pp. 2705-2705
Author(s):  
P. Rosenthal ◽  
A. Sołtysiak
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Joint Spectral Radius

Algebra cone generalized b-metric space over Banach algebra and fixed point theorems of generalized Lipschitz mappings

2018 ◽  
Vol 11 (05) ◽  
pp. 1850068
Author(s):  
Aziz Ahmed ◽  
J. N. Salunke
Keyword(s):  
Fixed Point ◽  
Banach Algebra ◽  
Spectral Radius ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Lipschitz Mappings

In this paper, we introduce the concept of algebra cone generalized [Formula: see text]-metric space over Banach algebra by replacing the constant [Formula: see text] with [Formula: see text] where [Formula: see text] is the spectral radius of [Formula: see text], with this modification we shall prove Banach and Kannan fixed point results for contractive generalized Lipschitz mappings in such a space. Moreover, we present one example in support of our results.

scholarly journals Characterizations of a commutative semisimple modular annihilator Banach algebra through its socle

2021 ◽  
Vol 40 (3) ◽  
pp. 697-709
Author(s):  
Youness Hadder ◽  
Abdelkhalek El Amrani
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Complex Semisimple ◽  
Semisimple Banach Algebra

Let A be a commutative complex semisimple Banach algebra. In this paper we continue the study of kh(soc(A)). Thus we will give, among other things, some new characterizations of this ideal in terms of the closure, in the spectral radius norm, of the socle of A.

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.

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