On a factorization of operators as a product of an essentially unitary operator and a strongly irreducible operator

As it is well-known, for any operator T on a complex separable Hilbert space, T   has the polar decomposition T=U|T|T=U|T|, where U   is a partial isometry and |T||T| is the non-negative operator (T⁎T)12. In this paper, we will give a decomposition theorem in a new sense that |T||T| will be replaced by a strongly irreducible operator. More precisely, for any operator T   and any ε>0ε>0, there exists a decomposition T=(U+K)ST=(U+K)S, where U is a partial isometry, K   is a compact operator with ||K||<ε||K||<ε and S is strongly irreducible.