Alternating direction method of multipliers (ADMM) is a powerful first order methods for various applications in signal processing and imaging. However, there is no clear result on the weak convergence of ADMM with relaxation studied by Eckstein and Bertsakas cite{EP} in infinite dimensional Hilbert spaces. In this paper, by employing a kind of partial gap analysis, we prove the weak convergence of general preconditioned and relaxed ADMM in infinite dimensional Hilbert spaces, with preconditioning for solving all the involved implicit equations under mild conditions. We also give the corresponding ergodic convergence rates respecting to the partial gap function. Furthermore, the connections between certain preconditioned and relaxed ADMM and the corresponding Douglas-Rachford splitting methods are also discussed, following the idea of Gabay in cite{DGBA}. Numerical tests also show the efficiency of the proposed overrelaxation variants of preconditioned ADMM.
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