On Wold-type decomposition

The main result of this paper is some new equivalent condition for a contraction to be a direct sum of a unitary operator and a C.0 completely nonunitary contraction (called Wold-type decomposition). It is applied for (p,k)-quasihyponormal, k*-paranormal, k-paranormal and of class Q operators. This decomposition is connected with the Putnam-Fuglede property. Additionally, we show an example of operator of class Q with norm 1, but without SVEP (so not paranormal). Finally, we remark that each k*-paranormal operator is (k+1)-paranormal.