On The Spectra And Numerical Ranges Of Multi-Commutative Operators

In this paper,the relation between spectra and numerical ranges of multi-operators on Banach spaces is studied,an open problem in the field of the non-linear control is partly answered. In addition,the representation of the Taylor's spectra of the Toeplitz operators on the Dirichlet space is discussed.In particular ,the Predholm spectra of multi-Toeplitz operators with symbols in H?D)+Cl(D) are given. The problem of the convexity of the Toeplitz operators on Dirichlet space is also discussed.