Similar operators

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 $.