## The essential norm of an operator and its adjoint

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- by Sheldon Axler, Nicholas Jewell and Allen Shields
- Trans. Amer. Math. Soc.
**261**(1980), 159-167 - DOI: https://doi.org/10.1090/S0002-9947-1980-0576869-9
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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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## Bibliographic Information

- © Copyright 1980 American Mathematical Society
- 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