Spectral measures for derivative powers via matrix-valued Clark theory

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...

The theory of matrix-valued multiresolution analysis frames

Spectral Factorization of Non-Rational Matrix-Valued Spectral Densities

Recently, a necessary and sufficient uniform log-integrability condition has been established for the canonical spectral factorization mapping to be sequentially continuous. Under this condition, if a sequence of spectral densities converge to a limiting spectral density then the canonical spectral factors of the sequence converges to the canonical spectral factor of the limiting density. Howev...

Spectral estimates for matrix-valued periodic Dirac operators

We consider the first order periodic systems perturbed by a 2N × 2N matrix-valued periodic potential on the real line. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov fun...

Inverse spectral analysis for finite matrix-valued Jacobi operators

Consider the Jacobi operators J given by (J y)n = anyn+1+bnyn+a∗n−1yn−1, yn ∈ C (here y0 = yp+1 = 0), where bn = b ∗ n and an : det an 6= 0 are the sequences of m × m matrices, n = 1, .., p. We study two cases: (i) an = a∗n > 0; (ii) an is a lower triangular matrix with real positive entries on the diagonal (the matrix J is (2m+1)-band mp×mp matrix with positive entries on the first and the las...