Operator-valued spectral measures and large deviations

Let H be a Hilbert space, U an unitary operator on H and K a cyclic subspace for U . The spectral measure of the pair (U,K) is an operator-valued measure μK on the unit circle T such that ∫ T zdμK(z) = ( PKU k ) ↾K , ∀ k ≥ 0 where PK 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 ([20]). 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. 1 ha l-0 08 70 27 6, v er si on 1 6 O ct 2 01 3