On Reflexivity of Certain Hyponormal Operators with Double Commutant Property

Sarason did pioneer work on the reflexivity and purpose of this paper is to discuss the reflexivity of different class of contractions. Among contractions it is now known that C11 contractions with finite defect indices, C.o contractions with unequal defect indices and C1. contractions with at least one finite defect indices are reflexive. More over the characterization of reflexive operators among co contractions and completely non unitary weak contractions with finite defect indices has been reduced to that of S (F), the compression of the shift on H2 ⊖ F H2, F is inner. The present work is mainly focused on the reflexivity of contractions whose characteristic function is constant. This class of operator include many other isometries, co-isometries and their direct sum. We shall also discuss the reflexivity of hyponormal contractions, reflexivity of C1. contractions and weak contractions. It is already known that normal operators isometries, quasinormal and sub-normal operators are reflexive. We partially generalize these results by showing that certain hyponormal operators with double commutant property are reflexive. In addition, reflexivity of operators which are direct sum of a unitary operator and C.o contractions with unequal defect indices,is proved Each of this kind of operator is reflexive and satisfies the double commutant property with some restrictions.

К расчету поля конечного магнитного цилиндра с внутренним соосным цилиндрическим дефектом

Using the grid method, a numerical solution of the direct problem of magnetostatics for calculating the field of a finite cylinder with a constant magnetic permeability containing an internal inclusion in the form of a coaxial cylinder with a different magnetic permeability is carried out. The algorithm is created for an arbitrary external field. In order to assess the reliability and accuracy of the solution method, the results were tested using precisely solved problems. A comparison is also made with the results of the previously solved problem of a finite defect-free cylinder. The coordinate dependences of the components of the resulting field are constructed for different source data. The program adds to the library of magnetic control problems and can be used for high-quality verification with the results of model experiments, as well as for evaluating the geometric characteristics of an internal defect.

Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators

In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invariant subspaces with a finite defect. This is a powerful result which allows us to gain insight in to the structure of scalar subspaces of the Hardy space using vector-valued Hardy space techniques. These results have far reaching applications, in particular they allow us to develop an all encompassing approach to the study of the kernels of: the Toeplitz operator, the truncated Toeplitz operator, the truncated Toeplitz operator on the multiband space and the dual truncated Toeplitz operator.