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Abstract
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A linear functional
on the
space of bounded sequences
is called a Banach limit if it is positive, normalised and invariant under the
shift operator. There are Banach limits which possess additional invariance
properties. We prove that every Banach limit invariant under the Cesaro
operator is also invariant under all dilation operators. We also prove the
“continuous version” of this result and apply it to the theory of singular
traces.
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Keywords
Banach limit, space of bounded sequences, Cesaro operator,
dilation operator
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Mathematical Subject Classification 2010
Primary: 46B45, 58B34
Secondary: 47B37
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Milestones
Received: 22 March 2018
Revised: 8 June 2019
Accepted: 7 December 2019
Published: 14 June 2020
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