Function calculus for almost isometric operators

Representations of some algebras of functions in the commutant of an almost isometric operator (i.e. a trace class perturbation of an isometry) are constructed. Properties of these representations are investigated. In particular, an analog of the class $C_0$ for contractions is discovered: it is shown that an operator is singular (i.e. the boundary values of its resolvent from inside and outside the disc coincide) if and only if there exists a nonzero function $\varphi$ for which $\varphi(T)=0$. Bibliography: 7 titles.