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For a Hilbert space operator T∈B(H), let LT and RT∈B(B(H)) denote, respectively, the operators of left multiplication and right multiplication by T. For positive integers m and n, let ▵T∗,Tm(I)=(LT∗RT−I)m(I) and δT∗,Tn(I)=(LT∗−RT)m(I). The operator T is said to be (m,n)-isosymmetric if ▵T∗,TmδT∗,Tn(I)=0. Power bounded (m,n)-isosymmetric operators T∈B(H) have an upper triangular matrix representation T=T1T30T2∈B(H1⊕H2) such that T1∈B(H1) is a C0.-operator which satisfies δT1∗,T1n(I|H1)=0 and T2∈B(H2) is a C1.-operator which satisfies AT2=(Vu⊕Vb)|H2A, A=limt→∞T2∗tT2t, Vu is a unitary and Vb is a bilateral shift. If, in particular, T is cohyponormal, then T is the direct sum of a unitary with a C00-contraction.

For a Hilbert space operator T ∈ B (H) , let L T and R T ∈ B (B (H)) denote, respectively, the operators of left multiplication and right multiplication by T. For positive integers m and n, let ▵ T ∗ , T m (I) = (L T ∗ R T − I) m (I) and δ T ∗ , T n (I) = (L T ∗ − R T) m (I) . The operator T is said to be (m , n) -isosymmetric if ▵ T ∗ , T m δ T ∗ , T n (I) = 0 . Power bounded (m , n) -isosymmetric operators T ∈ B (H) have an upper triangular matrix representation T = T 1 T 3 0 T 2 ∈ B (H 1 ⊕ H 2) such that T 1 ∈ B (H 1) is a C 0. -operator which satisfies δ T 1 ∗ , T 1 n (I | H 1) = 0 and T 2 ∈ B (H 2) is a C 1. -operator which satisfies A T 2 = (V u ⊕ V b) | H 2 A , A = lim t → ∞ T 2 ∗ t T 2 t , V u is a unitary and V b is a bilateral shift. If, in particular, T is cohyponormal, then T is the direct sum of a unitary with a C 00 -contraction. [ABSTRACT FROM AUTHOR]