Convergence of Associated Continued Fractions Revised

This is an expository article which contains alternative proofs of many theorems concerning convergence of a continued fraction to a holomorphic function. The continued fractions which are studied are continued fractions of the form where {a _n }, {b _n } are real sequences with a _n >0 (associated continued fractions). The proofs rely on the properties of the resolvent (–T)^–1, where T is the symmetric tridiagonal operator corresponding to {a _n } and {b _n }, and avoid most of technical aspects of earlier work. A variety of well-known results is proved in a unified way using operator methods. Many proofs can be regarded as functional analytic proofs of important classical theorems.