We present a generalization of Koplienko-Neidhardt trace formula for pairs of Hilbert space operators (T, V ) with T contractive and V unitary such that T − V is a Hilbert-Schmidt operator. We extend the result to pairs of contractions and then, via Cayley transform, to pairs of maximal dissipative operators. where ξ is an integrable real function that depends only on the pair (U 1 , U 0 ) and is determined by (0.1) up to an additive constant. The identity (0.1) is usually referred to as the trace formula. A function ξ, as in the trace formula, is called a spectral shift function of the pair (U 1 , U 0 ). A distinguished spectral shift function ξ 0 is fixed by the conditionwhere the operator value of Log U 1 U −1 0 is chosen so that the spectrum of 1/iLogM. Sh. Birman and M. G. Kreȋn [2] established that the pair (U 1 , U 0 ) admits a family of unitary scattering matrices {S(e it )} such that the identityholds for almost every t ∈ [−π, π] with e it belonging to the spectrum of U 0 . The spectral shift function was introduced by I. M. Lifshits [10] in connection with a problem of quantum statistic and for pairs of self-adjoint operators (for which a similar trace formula holds). After the pioneering work of M. G. , various interesting problems concerning (0.1) and (0.2) have attracted the attention
In this article we deal with the Arov-Grossman functional model to describe all the solutions of the Covariance Extension Problem for q-variate stationary stochastic processes and we find the density that maximizes the Burg Multivariate Entropy. This description is based on a one-to-one correspondence between the set of all solutions of the Covariance Extension Problem and the set of all contractive analytic functions H from the open unit disk with valueson the space of q × q matrices. With this correspondence, the density that maximizes the Burg Multivariate Entropy corresponds to the function H ≡ 0. Also, from the information that the Arov-Grossman functional model provides we obtain a version of the Levinson algorithm. The partial autocorrelation coefficient matrices are computed directly from Levinson's recursions.
A Schur-type analysis of the minimal weak unitary dilations of a given contraction operator is obtained from the Arov-Grossman functional model. The result is combined with the coupling method to give a description of the interpolants in the Relaxed Commutant Lifting Theorem. (2000). Primary 47A20; Secondary 47A57, 47A56.
Mathematics Subject Classification
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