Self-adjoint commuting ordinary differential operators

In this paper we study self-adjoint commuting ordinary differential operators of rank two. We find sufficient conditions when an operator of fourth order commuting with an operator of order 4g+2 is self-adjoint. We introduce an equation on potentials V(x),W(x) of the self-adjoint operator \(L=(\partial_{x}^{2}+V)^{2}+W\) and some additional data. With the help of this equation we find the first example of commuting differential operators of rank two corresponding to a spectral curve of higher genus. These operators have polynomial coefficients and define commutative subalgebras of the first Weyl algebra.     

Self-adjoint extensions of symmetric operators

Let ? denote the Hilbert space of analytic functions on the unit disk which are square summable with respect to the usual area measure. In this paper we consider the formal differential exepressons of order two or greater having the form {fx321-1} and {fx321-2} which give rise to symmetric operators in ?. We show that these operators in ? admit self-adjoint extensions in ?.     

Self-adjoint bound-preserving extensions of semibounded hermitian operators

It is shown by concrete examples that in the case of Hermitian (not densely defined) operators their self-adjoint bound-preserving extensions exist not always. A sufficient condition for the existence of such extensions is established.Translated from Dinamicheskie Sistemy, No. 4, pp. 120–125, 1985.     

Self-adjoint subspace extensions of nondensely defined symmetric operators

The self-adjoint subspace extensions of a possibly nondensely defined symmetric operator in a Hilbert space are characterized in terms of “generalized boundary conditions.”     

On the commuting extensions of subnormal operators

We give some results concerning the following problem: Given a linear bounded operatorA which is subnormal on a Hilbert spaceH, andB its minimal normal extension on a Hilbert spaceK ～H, when can a quasi-normal operatorT commuting withA be extended to an operatorT ^e onK such thatT ^e commutes withB andT ^e is quasi-normal onK?     

Self-adjoint curl operators

We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain D. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem, this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl, D) equipped with a symplectic pairing arising from the ${\wedge}$ -product of 1-forms on ${\partial D}$ . Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extensions. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed.     

Self-adjoint abstract differential operators

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H₁ whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula AE, B(x)0, and the operator B(x) is bounded for all x. An operator L₀ is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) H₁ according to the formula: L₀y=–y + Q(x)y (–0 is a self-adjoint operator in H₁ under the given assumptions.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 65–72, July, 1969.     

Inhomogeneous semigroups of commuting operators

The concepts of the homogeneously continuable semigroup of operators, and of infinitesimal and reproducing families of a semigroup, are introduced. The class of strongly continuous homogeneously continuable semigroups of commuting linear operators is discussed. This class contains in particular the class (C₀) of homogeneous semigroups. An analog of the Hill-Yosida theorem is proved for it.     

Self-adjoint extensions for second-order symmetric linear difference equations

In this paper, self-adjoint extensions for second-order symmetric linear difference equations with real coefficients are studied. By applying the Glazman-Krein-Naimark theory for Hermitian subspaces, both self-adjoint subspace extensions and self-adjoint operator extensions of the corresponding minimal subspaces are completely characterized in terms of boundary conditions, where the two endpoints may be regular or singular.     

Asymptotically commuting families of operators

We study families of symmetric operators (Q _n) with domains given by the range of self-adjoint contraction semigroups (e ^−tHn ). Assuming the asymptotic commutativity, lim _n [Q _n, e^−tHn]=0, and certain other estimates, we establish the existence and properties of a limiting self-adjoint operatorQ=lim _n Q _n. We apply these results to the study of an elementary supersymmetry algebra. Supported in part by the National Science Foundation under Grant DMS/PHY 88-16214.     

Convergence of Hermite interpolatory operators

Divergence-free wavelets play important roles in both partial differential equations and fluid mechanics. Many constructions of those wavelets depend usually on Hermite splines. We study several types of convergence of the related Hermite interpolatory operators in this paper. More precisely, the uniform convergence is firstly discussed in the second part; then, the third section provides the convergence in the Donoho’s sense. Based on these results, the last two parts are devoted to show the convergence in some Besov spaces, which concludes the completeness of Bittner and Urban’s expansions.