On the resolvent and the principal vectors of a compact linear operator
John R. Ringrose · Mathematical Proceedings of the Cambridge Philosophical Society · 1964
Let T be a compact linear operator acting in a complex Hilbert space H. Then there is a simple resolution of the identity {Eλ} in H which reduces T (for a proof of this statement, and for definitions of the terms used, see (l), section 5, especially Theorem 6). In the present paper we give constructions for the resolvent of T, and for a complete set of principal (that is, eigen and adjoined) vectors, in terms of T, {Eλ}, and the diagonal coefficients {αλ} of T. These constructions require no technique more involved than the expansion of a resolvent operator in a Neumann series.