Itis 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