Integr. equ. oper. theory 46 (2003) 125–164 0378-620X/03/020125-40 c  2003 Birkh¨auser Verlag Basel/Switzerland Integral Equations and Operator Theory A Bitangential Interpolation Problem on the Closed Unit Ball for Multipliers of the Arveson Space Joseph A. Ball and Vladimir Bolotnikov Abstract. We solve a bitangential interpolation problem for contractive mul- tipliers on the Arveson space with an arbitrary interpolating set in the closed unit ball B d of C d . The solvability criterion is established in terms of positive kernels. The set of all solutions is parametrized by a Redheffer transform. Mathematics Subject Classification (2000). Primary 47A57; Secondary 30E05. Keywords. Arveson space, multipliers, unitary extensions. 1. Introduction In this paper we study the bitangential interpolation problem for a class of con- tractive valued functions on the unit ball of C d . To introduce this class we first recall some definitions. Let Ω be a domain in C d , let E be a separable Hilbert space and let L(E ) stand for the set of all bounded linear operators on E .A L(E )–valued function K(z,w) defined on Ω × Ω is called a positive kernel if n  j,=1 c ∗ j K(z (j) ,z () )c  ≥ 0 for every choice of an integer n, of vectors c 1 ,...,c n ∈E and of points z (1) ,..., z (n) ∈ Ω. This property will be denoted by K(z,w)  0. In what follows we shall write K w (z) rather than K(z,w) if the last function will be considered as a function of z with a fixed point w ∈ Ω. For example, the kernel k d (z,w)= 1 1 −z, w C d (1.1)