Some results on eigenvalues of finite type, resolvents and Riesz projections

The present paper considers a separable Hilbert space and a bounded linear operator that has an eigenvalue of finite type at . Using the image and the kernel of certain operators that are defined recursively starting from , a construction of the local Smith form and extended canonical system of root functions of at is provided. This allows to express the partial multiplicities, the algebraic multiplicity, the Jordan structure and Jordan chains of A at in terms of the quantities defined by the recursion. Similarly, formulas for the rank, image and kernel of the operators in the principal part of the resolvent at , and thus also of the Riesz projection, are given in terms of the quantities defined by the recursion.