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On the Entropy Numbers of an Operator and its Adjoint

✍ Scribed by David E. Edmunds; Hans-Olav Tylli

Publisher
John Wiley and Sons
Year
1986
Tongue
English
Weight
463 KB
Volume
126
Category
Article
ISSN
0025-584X

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✦ Synopsis

The main purpose of this paper is to give some natural relations between the entropy numbers of an operator and those of its adjoint. This problem has attracted some recent attention (of. [ll], 14.3. 6 and[a]). Typically, we shall consider inequalities which allow a correction term. We obtain our first inequalities as a consequence of some simple and natural inequalities between the KOLMOCIOROV (and the GELFAND) numbers and the entropy numbers that deserve to be better known.

I n section 2 we estimate the entropy numbers of the Hom-product of a pair of operators by modifying some methods of I<. VALA [la, 151. By specializing, one obtains types of inequalities similar to those of 8 1. However, this section is of independent interest as the Hom-product is connected with tensor products of operators.

I n the final section we give an inequality dealing with the same question for the approximation numbers, together with remarks on the circle of ideas related to the well-known inequality of WEYL on the distribution of eigenvalues (see [ S ] ) , and also pose some further questions.

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