Complete norm-preserving extensions of holomorphic functions

We show that for every connected analytic subvariety V there is a pseudoconvex set Ω such that every bounded matrix-valued holomorphic function on V extends isometrically to Ω. We prove that if V is two analytic discs intersecting at one point, if every bounded scalar valued holomorphic function extends isometrically to Ω, then so does every matrix-valued function. In the special case that Ω is the symmetrized bidisc, we show that this cannot be done by finding a linear isometric extension from the functions that vanish at one point.