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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the ideals of strictly singular and inessential operators

Author: William Pfaffenberger
Journal: Proc. Amer. Math. Soc. 25 (1970), 603-607
MSC: Primary 47.45
MathSciNet review: 0264442
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Abstract: In the present paper, we prove that if $X$ is a subprojective Banach space, then the ideal of strictly singular operators on $X$ is equal to the ideal of inessential operators on $X$. We give an example to show that equality does not hold for all Banach spaces $X$. We also investigate the relationship between the semi-Fredholm operators on a Banach space and the right and left null divisors in the quotient algebra of all the bounded operators modulo the ideal of compact operators. We are able to get some complete characterizations of the null divisors when the Banach space is subprojective.

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Keywords: Strictly singular operators, inessential operators, compact operators, Jacobson radical, subprojective Banach space, semi-Fredholm operators
Article copyright: © Copyright 1970 American Mathematical Society