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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The essential norm of an operator and its adjoint


Authors: Sheldon Axler, Nicholas Jewell and Allen Shields
Journal: Trans. Amer. Math. Soc. 261 (1980), 159-167
MSC: Primary 47A30; Secondary 41A35
DOI: https://doi.org/10.1090/S0002-9947-1980-0576869-9
MathSciNet review: 576869
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Abstract: We consider the relationship between the essential norm of an operator T on a Banach space X and the essential norm of its adjoint $ T^{\ast}$. We show that these two quantities are not necessarily equal but that they are equivalent if $ X^{\ast}$ has the bounded approximation property. For an operator into the sequence space $ {c_0}$, we give a formula for the distance to the compact operators and show that this distance is attained. We introduce a property of a Banach space which is useful in showing that operators have closest compact approximants and investigate which Banach spaces have this property.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0576869-9
Article copyright: © Copyright 1980 American Mathematical Society