Abstract
It is a known fact in operator theory that two similar operators have equal spectra but they
do not necessarily have to belong to the same class of operators. However, under the
stronger relation of unitary equivalence it can be shown that two unitarily equivalent
operators may belong to the same class of operators. In this paper we endeavor to exhibit
some results on some pairs of operators which may belong to the same class under not only
unitary equivalence but also isometric and co-isometric equivalence.