Abstract. In this note, we discuss the problem of membership in the classes ${\msbm A}_{m,n}$ (subclasses of the set of absolutely continuous contractions on Hilbert space with isometric Sz. Nagy--Foias functional calculus) through concrete examples as well as related topics. In particular we prove that $S\oplus S^*\not\in{\msbm A}_{2,2}$, where $S$ denotes the unilateral shift of multiplicity one. Besides considering a series of examples, we introduce intermediate additional subclasses which make the characterization of the classes ${\msbm A}_{n,n}$ somewhat clearer. We also study the question of the effect on the membership in ${\msbm A}_{m,n}$ when certain direct summands are ``removed''. In particular we prove that if $A\in C_0$, then $T\oplus A\in{\msbm A}_{m,n}$ implies that $T\in{\msbm A}_{m,n}$.

AMS Subject Classification (1991): 47D27, 47A20, 47A15

Received August 4, 1997 and in revised form May 29, 1998. (Registered under 3318/2009.)