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.

GREEN'S FUNCTIONS OF SOME BOUNDARY VALUE PROBLEMS FOR BYHARMONIC OPERATORS AND THEIR CORRECT CONSTRICTIONS

In this paper, a constructive method is given for constructing the Green function of the Dirichlet problem for a biharmonic equation in a multidimensional ball. The need to study boundary value problems for elliptic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, thermal conductivity, elasticity theory, and quantum physics. The distributions of the potential of the electrostatic field are described using the Poisson equation. When studying the vibrations of thin plates of small deflections, biharmonic equations arise. There are various ways to construct the Green Function of the Dirichlet problem for the Poisson equation. For many types of domains, it is constructed explicitly. And for the Neumann problem in multidimensional domains, the construction of the Green function is an open problem. For the ball, the Green function of the internal and external Neumann problem is constructed explicitly only for the two-dimensional and three-dimensional cases. Finding general correct boundary value problems for differential equations is always an urgent problem. The abstract theory of operator contraction and expansion originates from the work of John von Neumann, in which a method for constructing self-adjoint extensions of a symmetric operator was described and a theory of extension of symmetric operators with finite defect indices was developed in detail. Many problems for partial differential equations lead to operators with infinite defect indices. In the early 80s of the last century, M.O. Otelbaev and his students built an abstract theory that allows us to describe all correct constrictions of a certain maximum operator and separately - all correct extensions of a certain minimum operator, independently of each other, in terms of the inverse operator. In this paper, the correct boundary value problems for the biharmonic operator are described using the Green's function.

Self-Adjoint Extension and Spectral Theory of a Linear Relation in a Hilbert Space

The aim of this paper is to develop the conditions for a symmetric relation in a Hilbert space ℋ to have self-adjoint extensions in terms of defect indices and discuss some spectral theory of such linear relation.

J-Self-Adjoint Extensions for a Class of Discrete Linear Hamiltonian Systems

This paper is concerned with formallyJ-self-adjoint discrete linear Hamiltonian systems on finite or infinite intervals. The minimal and maximal subspaces are characterized, and the defect indices of the minimal subspaces are discussed. All theJ-self-adjoint subspace extensions of the minimal subspace are completely characterized in terms of the square summable solutions and boundary conditions. As a consequence, characterizations of all theJ-self-adjoint subspace extensions are given in the limit point and limit circle cases.

Simplicity and Spectrum of Singular Hamiltonian Systems of Arbitrary Order

The paper is concerned with singular Hamiltonian systems of arbitrary order with arbitrary equal defect indices. It is proved that the minimal operator generated by the Hamiltonian system is simple. As a consequence, a sufficient condition is obtained for the continuous spectrum of every self-adjoint extension of the minimal operator to be empty in some interval and for the spectrum to be nowhere dense in this interval in terms of the numbers of linearly independent square integrable solutions.