Abstract. The objective of this paper is to study wandering subspaces for commuting tuples of bounded operators on Hilbert spaces. It is shown that, for a large class of analytic functional Hilbert spaces H K on the unit ball in C n , wandering subspaces for restrictions of the multiplication tuple M z = (M z1 , . . . , M zn ) can be described in terms of suitable H K -inner functions. We prove that H K -inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogenous polynomials as an application. Along the way we prove a refinement of a result of Arveson on the uniqueness of minimal dilations of pure row contractions.

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In this article, we study a class of contractive factors of\break m-hypercontractions for m∈N. We find a characterization of such factors and it is achieved by finding explicit dilations of these factors on certain weighted Bergman spaces. This is a generalization of the work done by {B.K. Das, S. Sarkar, J. Sarkar}, Factorization of contraction, \textit{Adv. in Math.} \textbf{322}(2017), 186--200.

In this article we study commutant lifting, more generally intertwining lifting, for different reproducing kernel Hilbert spaces over two domains in C n , namely the unit ball and the unit polydisc. The reproducing kernel Hilbert spaces we consider are mainly weighted Bergman spaces. Our commutant lifting results are explicit in nature and that is why these results are new even in one variable (n = 1) set up. Notation NThe set of all natural numbers. Z +The set of all positive integers. D Open unit disc in the complex plane C. D nOpen unit polydisc in C n . B nOpen unit ball in C n . H, EHilbert spaces. B(H)The space of all bounded linear operators on H. H 2 E (D) E-valued Hardy space on D. All Hilbert spaces are assumed to be over the complex numbers.

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