Application of symmetric analytic functions to spectra of linear operators

The paper is devoted to extension of the theory of symmetric analytic functions on Banach sequence spaces to the spaces of nuclear and $p$-nuclear operators on the Hilbert space. We introduced algebras of symmetric polynomials and analytic functions on spaces of $p$-nuclear operators, described algebraic bases of such algebras and found some connection with the Fredholm determinant of a nuclear operator. In addition, we considered cases of compact and bounded normal operators on the Hilbert space and discussed structures of symmetric polynomials on corresponding spaces.

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.

Non-negative Integer Power of a Hyponormal m-isometry is Reflexive

Sarason did pioneer work on reflexive operator and reflexivity of normal operators, however, he did not used the word reflexive but his results are equivalent to say that every normal operator is reflexive. The word reflexive was suggested by HALMOS and first appeared in H. Rajdavi and P. Rosenthals book `Invariant Subspaces’ in 1973. This line of research was continued by Deddens who showed that every isometry in B(H) is reflexive. R. Wogen has proved that `every quasi-normal operator is reflexive’. These results of Deddens, Sarason, Wogen are particular cases of theorem of Olin and Thomson which says that all sub-normal operators are reflexive. In other direction, Deddens and Fillmore characterized these operators acting on a finite dimensional space are reflexive. J. B. Conway and Dudziak generalized the result of reflexivity of normal, quasi-normal, sub-normal operators by proving the reflexivity of Vonneumann operators. In this paper we shall discuss the condition under which m-isometries operators turned to be reflexive.