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1980•Transactions of the American Mathematical SocietyOpen access

The Essential Norm of an Operator and its Adjoint

Sheldon Axler, Nicholas P. Jewell, Allen Shields

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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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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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Available 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.

Key concepts: Mathematics, Approximation property, Finite-rank operator, Compact operator, Banach space, Operator norm, Norm (philosophy), Bounded operator

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