Functional spectrum of contractions

In this paper, we introduce a new kind of spectrum for the C⋅0-class contractions. Since elements in this spectrum are functions, rather than numbers, we shall call it functional spectrum. Functional spectrum is a “large” closed subset of the Hardy space over the unit disk, and in many cases there is a canonical embedding of classical spectrum into functional spectrum. The study is carried out in the setting of the Hardy space over the bidisk H2(D2), on which every C⋅0-class contraction has a representation. A key tool is reduction operator. The reduction operator also gives rise to an equivalent statement of the Invariant Subspace Problem.