scholarly journals The Automatic Continuity of Linear Operators on Some Semi-Prime Banach Algebra

2015 ◽  
Vol 4 (2) ◽  
pp. 43
Author(s):  
Youssef Tidli
Keyword(s):  
Banach Algebra ◽  
Linear Operators ◽  
Automatic Continuity

scholarly journals Compact linear operators from an algebraic standpoint

1967 ◽  
Vol 8 (1) ◽  
pp. 41-49 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Linear Operators ◽  
Identity Operator ◽  
Algebra Structure ◽  
Natural Setting ◽  
Norm Topology ◽  
Bounded Linear ◽  
Underlying Space ◽  
Closed Subalgebra

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. Let t be an element of B(X), and let edenote the identity operator on X. Since the earliest days of the theory of Banach algebras, ithas been understood that the natural setting within which to study spectral properties of t is the Banach algebra B(X), or perhaps a closed subalgebra of B(X) containing t and e. The effective application of this method to a given class of operators depends upon first translating the data into terms involving only the Banach algebra structure of B(X) without reference to the underlying space X. In particular, the appropriate topology is the norm topology in B(X) given by the usual operator norm. Theorem 1 carries out this translation for the class of compact operators t. It is proved that if t is compact, then multiplication by t is a compact linear operator on the closed subalgebra of B(X) consisting of operators that commute with t.

scholarly journals Uniqueness of the maximal ideal of operators on the ℓp-sum of ℓ∞n (n ∈ ) for 1 < p < ∞

2016 ◽  
Vol 160 (3) ◽  
pp. 413-421 ◽  
Author(s):  
TOMASZ KANIA ◽  
NIELS JAKOB LAUSTSEN
Keyword(s):  
Banach Space ◽  
Recent Result ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Maximal Ideal ◽  
American Mathematical Society ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Unique Maximal Ideal

AbstractA recent result of Leung (Proceedings of the American Mathematical Society, 2015) states that the Banach algebra ℬ(X) of bounded, linear operators on the Banach space X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓ1 contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓp and X = (⊕n∈$\mathbb{N}$ ℓ1n)ℓp whenever p ∈ (1, ∞).

scholarly journals Automatic discontinuity of intertwining operators

1982 ◽  
Vol 25 (1) ◽  
pp. 49-54 ◽  
Author(s):  
Sandy Grabiner
Keyword(s):  
Banach Spaces ◽  
Linear Transformation ◽  
Linear Operators ◽  
Intertwining Operators ◽  
Continuous Linear ◽  
Automatic Continuity ◽  
Converse Problem ◽  
Simple Modification ◽  
Continuous Linear Operators

Throughout this paper, we suppose that T and R are continuous linear operators on the Banach spaces X and Y, respectively. One of the basic problems in the theory of automatic continuity is the determination of conditions under which a linear transformation S: X → Y which satisfies RS = ST is continuous or is discontinuous. Johnson and Sinclair [4], [6], [11; pp. 24–30] have given a variety of conditions on R and T which guarantee that all such S are automatically continuous. In this paper we consider the converse problem and find conditions on the range S(X) which guarantee that S is automatically discontinuous. The construction of such automatically discontinuous S is then accomplished by a simple modification of a technique of Sinclair's [10; pp. 260–261], [11; pp. 21–23].

scholarly journals Automatic continuity for Banach algebras with finite-dimensional radical

2006 ◽  
Vol 81 (2) ◽  
pp. 279-296 ◽  
Author(s):  
Hung Le Pham
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Automatic Continuity ◽  
Algebraic Condition ◽  
Finite Dimensional ◽  
Complete Algebra

AbstractThe paper [3] proved a necessary algebraic condition for a Banach algebra A with finite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where A is separable and A/R is commutative, but is shown to fail in general. Similar questions for derivations are discussed.

Automatic Continuity for Linear Functions Intertwining Continuous Linear Operators on Frechet Spaces

1978 ◽  
Vol 30 (03) ◽  
pp. 518-530 ◽  
Author(s):  
Marc P. Thomas
Keyword(s):  
Banach Spaces ◽  
Functional Calculus ◽  
Linear Operators ◽  
Linear Functions ◽  
General Situation ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Automatic Continuity ◽  
Continuity Theory ◽  
Continuous Linear Operators

Many results concerning the automatic continuity of linear functions intertwining continuous linear operators on Banach spaces have been obtained, chiefly by B. E. Johnson and A. M. Sinclair [1; 2; 3; 5]. The purpose of this paper is essentially to extend this automatic continuity theory to the situation of Fréchet spaces. Our motive is partly to be able to handle the more general situation, since for example, questions about Fréchet spaces and LF spaces arise in connection with the functional calculus.

Automatic continuity of generalized local linear operators

1980 ◽  
Vol 32 (3-4) ◽  
pp. 263-294 ◽  
Author(s):  
Ernst Albrecht ◽  
Michael Neumann
Keyword(s):  
Linear Operators ◽  
Automatic Continuity ◽  
Local Linear

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.

Automatic Continuity of Homomorphisms in Non-associative Banach Algebras

2013 ◽  
Vol 65 (5) ◽  
pp. 989-1004
Author(s):  
C-H. Chu ◽  
M. V. Velasco
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Surjective Homomorphism ◽  
Automatic Continuity ◽  
Normed Algebra ◽  
Associative Algebras ◽  
Rare Element ◽  
Rare Elements

AbstractWe introduce the concept of a rare element in a non-associative normed algebra and show that the existence of such an element is the only obstruction to continuity of a surjective homomorphism from a non-associative Banach algebra to a unital normed algebra with simple completion. Unital associative algebras do not admit any rare elements, and hence automatic continuity holds.

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