Operator-valued spectral measures and large deviations

Let H be a Hilbert space, let U be a unitary operator on H and let K be a cyclic subspace for U. The spectral measure of the pair ( U , K ) is an operator-valued measure μ K on the unit circle T such that ∫ T z k d μ K ( z ) = ( P K U k ) ↾ K , ∀ k ≥ 0 where P K and ↾ K are the projection and restriction on K , respectively. When K is one dimensional, μ is a scalar probability measure. In this case, if U is picked at random from the unitary group U ( N ) under the Haar measure, then any fixed K is almost surely cyclic for U. Let μ ( N ) be the random spectral (scalar) measure of ( U , K ) . The sequence ( μ ( N ) ) was studied extensively, in the regime of large N. It converges to the Haar measure λ on T and satisfies the Large Deviation Principle at scale N with a good rate function which is the reverse Kullback information with respect to λ (Gamboa and Rouault, 2010). The purpose of the present paper is to give an extension of this result for general K (of fixed finite dimension p) and eventually to offer a projective statement (all p simultaneously), at the level of operator-valued spectral measures in infinite dimensional spaces.