Abstract. The first part is devoted to establish a useful mapping formula for functional calculus associated with $\rho $-contractions. In the second part, we give a general Julia's lemma for operators whose spectrum is contained in the closed unit disc. Finally, some applications, which are concerned with the hyperbolic metric on the Harnack parts of $\rho $-contractions, are given.

AMS Subject Classification (1991): 47A63, 47A15; 32D15, 47A11

Keyword(s): Integral formula for functional calculus, Julia's lemma for operators, \rho, -kernel, invariant subspaces

Received July 20, 2007, and in revised form July 10, 2008. (Registered under 6046/2009.)