Lie algebras of vector fields and differential operators on vector bundles

We study Lie algebras of vector fields on C manifolds endowed with an additional structure and also morphisms of bundles over such manifolds. Bibliography: 8 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 47–51.     

A criterion for the logarithmic differential operators to be generated by vector fields

We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give

• a sufficient criterion for the property,
• a simple proof of F.J. Calderón-Moreno's theorem that free divisors have the property,
• a proof that divisors in dimension with only isolated quasi-homogeneous singularities have the property,
• an example of a nonfree divisor with nonisolated singularity having the property,
• an example of a divisor not having the property, and
• an algorithm to compute the V-filtration along a divisor up to a given order.

     

Covariant differential operators

Supported in part by the Louisiana Education Quality Support Fund 86-LBR-016-04     

Commuting differential operators

We consider a model problem, related to Bessel functions, in the theory of commutative rings of linear differential operators with one independent variable. We construct new series of such commutative rings and give simple examples of rings with three generators.     

Traces of differential operators

The present survey is dedicated to the history of the state of the art in the theory of regularized traces of linear differential operators. Attention is mainly paid to operators with discrete spectrum.     

Skew differential operator algebras of twisted Hopf algebras

By introducing a twisted Hopf algebra we unify several important objects of study. Skew derivations of such an algebra are defined and the corresponding skew differential operator algebras are studied. This generalizes results in the Weyl algebra. Applying this investigation to the twisted Ringel-Hall algebra we get, in particular, a natural realization of the non-positive part of a quantized generalized Kac-Moody algebra, by identifying the canonical generators with some linear, skew differential operators. This also induces some algebras which are quantum-group like.     

Hypoelliptic ordinary differential operators

Necessary and sufficient conditions for the hypoellipticity of an ordinary differential operator withC ^∞ coefficients in a neighborhood of a zero of finite order of the leading term are given. A sufficient condition for such an operator to be in a certain Hörmander class is also given.     

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.     

Hankel operators that commute with second-order differential operators

Suppose that Γ is a continuous and self-adjoint Hankel operator on L²(0,∞) with kernel ?(x+y) and that with a(0)=0. If a and b are both quadratic, hyperbolic or trigonometric functions, and ? satisfies a suitable form of Gauss's hypergeometric differential equation, or the confluent hypergeometric equation, then ΓL=LΓ. The paper catalogues the commuting pairs Γ and L, including important cases in random matrix theory. There are also results proving rapid decay of the singular numbers of Hankel integral operators with kernels that are analytic and of exponential decay in the right half-plane.     

Spectrum of quasielliptic differential operators

Translated from Matematicheskie Zametiki, Vol. 53, No. 6, pp. 145–148, June, 1993.     

Supercyclic sequences of differential operators

Summary Several necessary and sufficient conditions for a sequence of infinite order differential linear operators on spaces of holomorphic functions on a domain of the complex plane to be supercyclic or c-hypercyclic are given in this paper, so completing earlier work of the authors on hypercyclicity, which in turn extended Birkhoff--MacLane--Godefroy--Shapiro's theorems. A new, general eigenvalue criterium for supercyclicity is also provided.     

Comparison of linear differential operators

In this paper, we study the question of the existence of the inequality $$\left\| {Q(D)f} \right\|_{L_q } \leqslant \gamma _0 \left\| {P(D)f} \right\|_{L_p } $$ , where P and Q are algebraic polynomials, D = d/dx, and γ ₀ is independent of the function f. We obtain criteria (necessary and simultaneously sufficient conditions) for the existence of such inequalities for functions on the circle, on the whole line, and on the semiaxis. Besides, for the semiaxis, we obtain an inequality for q = ∞ and any p ≥ 1 with the smallest constant γ ₀.     

Natural operators on tensor fields

Natural operators on tensor fields of different types are classified. We abstain from any assumptions such as symmetry or additional structures and consider arbitrary tensor fields.