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 vertical bar T vertical bar, where U is a partial isometry and vertical bar T vertical bar is the non-negative operator (T*T)(1/2). In this paper, we will give a decomposition theorem in a new sense that vertical bar T vertical bar will be replaced by a strongly irreducible operator. More precisely, for any operator T and any epsilon > 0, there exists a decomposition T = (U+K)S, where U is a partial isometry, K is a compact operator with parallel to K parallel to < epsilon and S is strongly irreducible. (C) 2015 Elsevier Inc. All rights reserved.