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

• DOI: 10.1016/j.jmaa.2015.03.052
• Publication Date: 2015-04-07T13:33:10Z
• Authors: Geng Tian; Yang Cao; Youqing Ji; Juexian Li
• Keywords: Operator (biology); Displacement operator; Isometry (Riemannian geometry); Weierstrass factorization theorem; Polar decomposition

Abstract As it is well-known, for any operator T on a complex separable Hilbert space, T has the polar decomposition T = U | T | , where U is a partial isometry and | T | is the non-negative operator ( T ⁎ T ) 1 2 . In this paper, we will give a decomposition theorem in a new sense that | T | will be replaced by a strongly irreducible operator. More precisely, for any operator T and any e > 0 , there exists a decomposition T = ( U + K ) S , where U is a partial isometry, K is a compact operator with | | K | | e and S is strongly irreducible.