Weylness Of Partial Operator Matrices

This dissertation deals with left(right)Weyl and Weyl completion problems of bounded operator matrices in Hilbert spaces,based on the perturbation property of semi-Fredholm operators,the characterization of non-compact operators and the space decom-position method.As applications,some complete descriptions of perturbation of the left(right)Weyl,left(right)essential,Weyl and essential spectra for Hamiltonian operator matrices are given,respectively.First,given the(bounded)operators A,B and C,we write Mx:=((?)).A nec-essary and sufficient condition is given for Mx to be a left Weyl(right Weyl or Weyl)operator for some operator X.Some relevant properties,including the connections be-tween the left(right)Weylness of Mx and the semi-Fredholmness of its known entries,are discussed in detail.Besides,some applications to the spectrum assignment problem in systems theory and an illustrating example with non-compact operator C are also given.Next,given the operators A and C,we define MX,Y:?((?)).A necessary and sufficient condition is given for MX,Y to be a left Weyl(right Weyl or Weyl)operator for some operators X and Y.In particular,we also obtain the necessary and sufficient condition for MX,Y to be a left Weyl(right Weyl,Weyl,left Fredholm,right Fredholm or Fredholm)operator for some operator X and self-adjoint operator Y.Then,for the operator matrix MX:=((?)),with given the diagonal entries A and B,we consider the self-adjoint perturbations of the spectra.A necessary and sufficient condition is given under which such operator matrix admits a left Weyl(right Weyl,left Fredholm,right Fredholm,Weyl or Fredholm)operator completion by choosing some bounded self-adjoint operator.It is shown that the self-adjoint perturbation of the left Weyl(right Weyl,left essential,right essential,Weyl or essential)spectrum can be a proper set of the general perturbation.Combining the spectral properties,we further characterize the perturbation of the left Weyl(right Weyl,left essential,right essential,Weyl or essential)spectrum for upper triangular Hamiltonian operator matrices.(?)Finally,given the diagonal entries A,B and C,we write MD,E,F =((?)).A characterization of pertubation of the spectrum for MD,E,F is given.As a byproduct,a necessary and sufficient condition under which the spectrum of MD,E,F is equal to the union of the spectrum of diagonal entries A,B and C is given.Moreover,we deals with general n×n upper-triangular operator matrices with given diagonal entries.Some com-plete descriptions of perturbations of their point spectra,residual spectra and continuous spectra are given,based on the space decomposition method.