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Let A : X ! X be a bounded operator on a separable complex Hilbert space X with an inner producth ;i X. For b;c2 X, a weak resolvent of A is the complex function of the formh(I zA) 1 b;ciX . We will discuss an equivalent condition, in terms of weak resolvents, for A to be similar to a restriction of the backward shift of multi- plicity 1. 1. Introduction. We characterize an operator on a separable complex Hilbert space which is similar to a part of the backward shift of multiplicity 1, in terms of cyclic vectors and weak resolvents belonging to the Hardy spaceH 1 . We base our development on the work of Fuhrmann (6), Helton (7),

Operators on a Hilbert space similar to a part of the backward shift of multiplicity one