The Spectra And Algebra Index Of Unitary Self-Adjoint Operator

This thesis mainly studies the spectrum and algebraic index of unitary self-adjoint operators.Firstly, a relationship between spectra for unitary self-adjoint operators and spectra for their adjoint is considered, it is shown that the point spectrum and the residual spectrum of unitary self-adjoint operators are equal to the point spectrum and the residual spectrum of their adjoint; the union of first (second) kind of point spectrum and the first (second) kind of residual spectrum, the third kind of point spectrum and the forth kind of point spectrum are, respectively, symmetric with respect to the real axis. At that time, the necessary and sufficient condition for the first (second) kind of residual spectrum to be empty set is that the first (second) kind of point spectrum is symmetric with respect to the real axis. Next, the range of the point spectrum, the residual spectrum for nonnegative unitary self-adjoint operator and the first kind of point spectrum for upper triangular unitary self-adjoint operator are characterized, respectively. Lastly, the sufficient conditions for the algebraic index of a class of unitary self-adjoint operator to be 1 are given. In addition, some examples are given to illustrate the correctness of our results.