Contractivity results for alternating direction schemes in Hilbert spaces

Abstract In this paper we obtain some contractivity results for operators R (− A 1 ,…,− A n ), where R ( z 1 ,…, z n ) are rational approximations to exp( z 1 +⋯+ z n ), and A i are maximal monotone operators on a Hilbert space H . A general result is proved by using an extension to several variables of a result of Von Neumann for bounding f ( A )( f a holomorphic function, A an operator on H ). This theory is applied to the convergence analysis of Alternating Direction methods and, more generally, to Fractional Steps schemes.