Contractions with rank one defect operators and truncated CMV matrices

The main issue we address in the present paper are the new models for completely nonunitary contractions with rank one defect operators acting on some Hilbert space of dimension N?∞. These models complement nicely the well-known models of Livšic and Sz.-Nagy-Foias. We show that each such operator acting on some finite-dimensional (respectively, separable infinite-dimensional Hilbert space) is unitarily equivalent to some finite (respectively semi-infinite) truncated CMV matrix obtained from the “full” CMV matrix by deleting the first row and the first column, and acting in CN (respectively ?²(N)). This result can be viewed as a nonunitary version of the famous characterization of unitary operators with a simple spectrum due to Cantero, Moral and Velázquez, as well as an analog for contraction operators of the result from [Yu. Arlinski?, E. Tsekanovski?, Non-self-adjoint Jacobi matrices with a rank-one imaginary part, J. Funct. Anal. 241 (2006) 383-438] concerning dissipative non-self-adjoint operators with a rank one imaginary part. It is shown that another functional model for contractions with rank one defect operators takes the form of the compression f(ζ)→PK(ζf(ζ)) on the Hilbert space L²(T,dμ) with a probability measure μ onto the subspace K=L²(T,dμ)?C. The relationship between characteristic functions of sub-matrices of the truncated CMV matrix with rank one defect operators and the corresponding Schur iterates is established. We develop direct and inverse spectral analysis for finite and semi-infinite truncated CMV matrices. In particular, we study the problem of reconstruction of such matrices from their spectrum or the mixed spectral data involving Schur parameters. It is pointed out that if the mixed spectral data contains zero eigenvalue, then no solution, unique solution or infinitely many solutions may occur in the inverse problem for truncated CMV matrices. The uniqueness theorem for recovered truncated CMV matrix from the given mixed spectral data is established. In this part the paper is closely related to the results of Hochstadt and Gesztesy-Simon obtained for finite self-adjoint Jacobi matrices.     

Functions of operators under perturbations of class Sp

This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α_‖ in terms of ‖A−BSp_‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K).     

Operator Hölder-Zygmund functions

It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to Hölder classes. Namely, we prove that if f belongs to the Hölder class Λα(R) with 0<α<1, then for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ₁(R): for arbitrary self-adjoint operators A and K we have . We also obtain analogs of this result for all Hölder-Zygmund classes Λα(R), α>0. Then we find a sharp estimate for ‖f(A)−f(B)‖ for functions f of class for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which for self-adjoint A and B, and for an arbitrary function f in Λω. We obtain similar estimates for commutators f(A)Q−Qf(A) and quasicommutators f(A)Q−Qf(B). Finally, we estimate the norms of finite differences for f in the class Λω,m that is defined in terms of finite differences and a modulus continuity ω of order m. We also obtain similar results for unitary operators and for contractions.     

Formal Deformations,Contractions and Moduli Spaces of Lie Algebras

Jump deformations and contractions of Lie algebras are inverse concepts, but the approaches to their computations are quite different. In this paper, we contrast the two approaches, showing how to compute the jump deformations from the miniversal deformation of a Lie algebra, and thus arrive at the contractions. We also compute contractions directly. We use the moduli spaces of real 3-dimensional and complex 3 and 4-dimensional Lie algebras as models for explaining a deformation theory approach to computation of contractions. The research of the authors was partially supported by grants from the Mathematisches Forschungsinstitut Oberwolfach, OTKA T043641, T043034 and the University of Wisconsin-Eau Claire.     

Absolute continuity of spectral shift

In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectral measure of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.     

Quasi-Classical Lie Algebras and their Contractions

After classifying indecomposable quasi-classical Lie algebras in low dimension, and showing the existence of non-reductive stable quasi-classical Lie algebras, we focus on the problem of obtaining sufficient conditions for a quasi-classical Lie algebras to be the contraction of another quasi-classical algebra. It is illustrated how this allows to recover the Yang-Mills equations of a contraction by a limiting process, and how the contractions of an algebra may generate a parameterized families of Lagrangians for pairwise non-isomorphic Lie algebras. Work partially supported by the research project MTM2006-09152 of the M.E.C.     

Contractive maps on normed linear spaces and their applications to nonlinear matrix equations

In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = I_n + A^∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A.     

Invariant functions and contractions of certain types of Lie algebras of lower dimensions

In this paper, we deal with contractions of Lie algebras. We use two invariant functions of Lie algebras as a tool, named ψ and ? function, respectively, which have a great application in Physics due to their remarkable properties. We focus the study of these functions in the frame of the filiform Lie algebras, trying to extend to these algebras some of the properties of such functions over semi-simple Lie algebras.