Unconditional Bases, the Matrix Muckenhoupt Condition, and Carleson Series in a Spectrum

Two families of functions constructed by a system of n scalar Muckenhoupt weights are studied. Criteria are given under which these families are unconditional bases. From the point of view of the spectral operator theory, the problem is reduced to the study of the structure of n-dimensional perturbations of the integration operator. Weighted estimates for the Hilbert transform in the spaces of vector-functions are applied to construct an operator mapping functions of the studied families to vector-valued rational functions. The concept of the Carleson series is used in the study of the following problem: when do vector-valued rational functions form an unconditional basis? Bibliography: 8 titles.     

Spectral analysis of biorthogonal expansions generated by Muckenhoupt weights

An arbitrary Muckenhoupt A₂-weight w² on the special contour ( 1/2) generates a function Y_,w (, t), which for =1, w²(z) 1 coincides with the exponential exp{it}. In the paper, with the aid of B. S. Pavlov's geometric approach, one obtains criteria for the unconditional basis property of families of functions of the form {y,w(_k,t):_k} in the space L₂(0, ). The analytic foundation of the constructions is a generalization of M. M. Dzhrbashyan's certain results (power weight) to the case of arbitrary Muckenhoupt A₂- weights.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 34–80, 1991.     

On the similarity to self-adjoint operators

An approach to the similarity problem is presented, which is based on the notion of a w-perturbation of the Volterra operator and uses the theory of Muckenhoupt weights.     

Representability of de Branges Matrices as Blaschke--Potapov Products and Completeness of Some Families of Functions

Criteria for the representability of meromorphic second-order matrix functions J-expanding in the upper half-plane (de Branges matrices) as left, right, and two-sided Blaschke--Potapov products are stated. Results on the spectral structure of operators whose characteristic matrix functions are de Branges matrices are obtained.     

On a class of completely continuous operators in Hilbert spaces

We introduce a class G of completely continuous operators and prove theorems on the spectral structure of these operators. In particular, operators of this class are similar to model operators in de Branges spaces.