Characterization of Singular Integral Operators and Inverse Spectral Theory for Symmetric Operators

Abstract

We investigate the problem of simultaneous diagonalization of the multiplication operators on Hilbert space modulo norm ideals of compact operates. We show that within a gap of the minimal Laplacian on any bounded domain in the Euclidean space each kind of absolutely continuous spectrum can be generated by a self- adjoint realization of the Laplacian on the bounded domain and in addition give results on mixed types of spectra, that is, absolutely continuous, singular continuous and point spectrum. We will work within the much more general framework of self-adjoint extensions of so called significantly deficient operators. We shall investigate the question about what spectral properties the self-adjoint extensions of the symmetric operator in a separable Hilbert space can have inside the gap and provide methods on how to construct self-adjoint extensions of the symmetric operator with prescribed spectral properties inside the gap. We characterize integral operators with semi–separable kernels in certain class that have different symmetries. We treat the self-adjoint case, the positive case, the J-unitary case. The positive real case, the dissipative case and the contractive case. We construct a family of reflexive Banch spaces with long transfinite bases but with no unconditional basic sequences. The method we introduce to a achieve this allows us to considerably control the structure of subspaces of the resulting spaces as well as to describe the corresponding spaces on non- strictly singular operators. We find subspaces such that the operator space is quite rich but any bounded operator is a strictly singular perturbation of as scalar multiple of the identity. We discuss rank one perturbation, and show the generalization results known for boundary condition dependence of strum Liouville operators on half-lines to the abstract rank one case. It is shown that for any self-adjoint extension of the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non-zero deficiency indices, the absolutely continuous spectrums of self-adjoin extensions are contained in that one of a self-adjoint extension. More-over for the classes of extensions the absolutely continuous parts of a self- adjoin extension and the orthogonal sum of infinitely many pairwise unitary equivalent symmetric operators with non-zero deficiency indices are even unitarily equivalent. We give a necessary and sufficient condition for a commuting tuple of self-adjoin operator to simultaneously diagonalizable modulo Schatten p-class. We show a very general obstruction result for the problem of simultaneous diagonalization of commuting tuples of self-adjoint operators and singular integral operators modulo norm ideals satisfying a quantitative variant of kuroda’s condition.