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Similar operators

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Operators $ S $ and $ T $( not necessarily bounded) on a Banach space $ X $ for which there exists a bounded operator $ U $ on $ X $ having a bounded inverse and such that the following relation applies:

$$ S = U ^ {-} 1 TU. $$

If $ U $ is a unitary operator, then $ S $ and $ T $ are said to be unitarily equivalent.

This concept is an example of the concept of similar mappings. Let $ f $ and $ g $ be two mappings of a set $ X $ into itself. If there is a bijection $ U: X \rightarrow X $ such that $ Uf = gU $, then these mappings are said to be similar. Attempts have been made to give a definition of similarity for mappings from one set $ X $ into another $ Y $; for example, such mappings are called similar if there exist bijections $ U $ and $ V $ of the sets $ X $ and $ Y $ into themselves such that $ Vf = gU $.

How to Cite This Entry:
Similar operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similar_operators&oldid=48699
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article