Canonical models and the self-adjoint parts of dissipative operators

Abstract

Let H be a complex separable Hilbert space and let A be a bounded operator on H with nonnegative imaginary part. The spectral invariants of the self-adjoint part Asa of A are described in terms of Livšic-Brodskiĭ characteristic functions of restrictions A∗ ¦ M, where M ranges over a suitably large collection of subspaces invariant for A∗. In case A = B + K, with B a self-adjoint multiplication operator and K a compact subdiagonal integral operator acting on a direct integral space, the multiplicity function of Asa is described in terms of properties of B. An application is given to Livšic's theory of triangular models