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Structural Properties of Elementary Operators

Published online by Cambridge University Press:  20 November 2018

Constantin Apostol  and
Lawrence Fialkow
Constantin Apostol
Affiliation:
Arizona State University, Tempe, Arizona
Lawrence Fialkow
Affiliation:
Arizona State University, Tempe, Arizona
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Let and denote complex Banach algebras and let b e a left Banach module and a right Banach -module. If

we define the bounded linear elementary operator R(A, B), acting on , by

For the case , elementary operators were introduced by Lumer and Rosenblum [19], who studied their spectral properties. In this setting many authors subsequently studied spectral, algebraic, metric, and structural properties of elementary operators, with particular attention devoted to the inner derivations δa (δa(x) = ax – xa) [25], generalized derivations τ(a, b) (τ(a, b)(x) = ax – xb) [9, 10], and elementary multiplications S(a, b) (S(a, b)(x) = axb), including left and right multiplications La and Rb [11].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 6 , 01 December 1986 , pp. 1485 - 1524
Copyright
Copyright © Canadian Mathematical Society 1986

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