Spectrum of compression of the coordinate multiplier

Let H be a reproducing kernel Hilbert space of analytic functions over the unit disk D . Let T z : f ↦ z f be the coordinate multiplier on H . Suppose that A is a zero set for H and I A is the invariant subspace for T z determined by A . Under some mild conditions, we prove that the spectrum of the compression of T z on H ⊖ I A is the closure of A . This covers the corresponding result in many classical function spaces. We give a uniform approach which does not rely on specific properties of a specific space.