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On the Operator Identity ∑ AkXBk ≡ 0

Published online by Cambridge University Press:  20 November 2018

C. K. Fong  and
A. R. Sourour
C. K. Fong
Affiliation:
University of Toronto, Toronto, Ontario
A. R. Sourour
Affiliation:
University of Victoria, Victoria, British Columbia
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Let Aj and Bj (1 ≦ j ≦ m) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by

(1)

We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the form

For example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 4 , 01 August 1979 , pp. 845 - 857
Copyright
Copyright © Canadian Mathematical Society 1978

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