Let Jgf(z)=∫10f(tz)Rg(tz)dtt be weighted Cesaro operator with holomorphic symbol g,and Igf(z)=∫10g(tz)Rf(tz)dtt be adjoint operator of Jg.

In this paper, based on the invariant subspace theory and adjoint operator concept of linear operator, a new matrix representation method is proposed to calculate the normal forms of n order general nonlinear dynamic systems.

First we prove that 0 is an eigenvalue of the operator with geometric multiplicity one,next we prove that all points on the imaginary axis except for zero belong to the resolvent set of the operator,last we prove that 0 is an eigenvalue of the adjoint operator of the operator.

We discuss the continuity of conjugate operator from L1wice conditon on a class of generalized Orlicz spaces L(M-1),2π).

In addition,we prove a lgebraic multiplicity of 0 for 1 and solving conjugate operator of system operator.

Gives the characterization of conjugate operators in conjugate spaces,proves a relation between an operator T and its double conjugate operator T,illustrates that the strongly irreducible property of an operator is not conjugate symmetric.

In this paper,it was given that the conditions for a 2×2 block matrix of operators L=AB C?can be completed into a invertible self adjoint operator with L -1 =** *D.

An improved adjoint operator method was proposed to compute the third order normal form of six dimensional nonlinear dynamical systems and the associated nonlinear transformation for the first time.

An improved adjoint operator method is employed to compute the third order normal form of six dimensional nonlinear dynamical systems and the associated nonlinear transformation for the first time.