A spectral mapping theorem for generating operators in second-order linear differential equations

For a class of second-order Cauchy problems, certain integrals of the fundamental solution operators are formally functions of a generating operator. We show that spectral mapping holds.     

Interpolation theorem for linear operators

In generalizing a series of known results, the following theorem is proved: If K is a continuous linear operator mapping E₀ into F₀ and E₁ into F₁ (where E₀, E₁ and F₀, F₁, being ideal spaces, are Banach lattices of functions defined on ₁ and ₂ respectively), then for any . (0, 1) K maps E ₀ ^1– E ₁ into [(F ₀ )^1–(F ₁ )] and is continuous; for suitably chosen norms in the spacesE ₀ ^1– E ₁ and [(F ₀ )^1–(F ₁ )] the norm of K is a logarithmically convex function of . Six titles are cited in the bibliography.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 593–598, December, 1967.The author wishes to thank M. A. Krasnosel'skii, under whose direction he is working.     

On Cartan's theorem for linear operators

In this paper, we describe a second main theorem of holomorphic curves in , of hyper‐order strictly less than 1, that involves a general linear operator . As an application, we derive a truncated second main theorem of degenerate holomorphic curves of hyper‐order strictly less than 1 using Nochka weights.     

Spline monotone approximation with linear differential operators

Let f∈C³[a,b] and L be a linear differential operator such that L(f)≥0. Then there exists a sequence Q_n, n≥1, of polynomial splines with equally spaced knots, such that Q^(r), approximates f^(r), 0≤r≤s, simultaneously in the uniform norm. This approximation is given through inequalities with rates, involving a measure of smoothness to f^(s); so that L (Q_n)≥0. The encountered cases are the continuous, periodic and discrete.     

A vanishing theorem for differential operators in positive characteristic

Let X be a smooth variety over an algebraically closed field k of characteristic p, and let F: X → X be the Frobenius morphism. We prove that if X is an incidence variety (a partial flag variety in type A _n ) or a smooth quadric (in this case p is supposed to be odd) then Hⁱ( X,End( \sfF_*O_X ) ) = 0 {H^i}\left( {X,\mathcal{E}nd\left( {{\sf{F}_*}{\mathcal{O}_X}} \right)} \right) = 0 for i > 0. Using this vanishing result and the derived localization theorem for crystalline differential operators [3], we show that the Frobenius direct image \sfF_*O_X {\sf{F}_*}{\mathcal{O}_X} is a tilting bundle on these varieties provided that p > h, the Coxeter number of the corresponding group.     

A non-existence theorem of lacunas for hyperbolic differential operators with constant coefficients

We prove a theorem on non-existence of lacunas of the fundamental solution for hyperbolic differential operators with constant coefficients.     

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.     

A continuum of unusual self-adjoint linear partial differential operators

In an earlier publication a linear operator _THar$T_{\mathrm{Har}}$ was defined as an unusual self-adjoint extension generated by each linear elliptic partial differential expression, satisfying suitable conditions on a bounded region Ω$\Omega$ of some Euclidean space. In this present work the authors define an extensive class of _THar$T_{\mathrm{Har}}$-like self-adjoint operators on the Hilbert function space _L2(Ω);$L_2(\Omega );$ but here for brevity we restrict the development to the classical Laplacian differential expression, with Ω$\Omega$ now the planar unit disk. It is demonstrated that there exists a non-denumerable set of such _THar$T_{\mathrm{Har}}$-like operators (each a self-adjoint extension generated by the Laplacian), each of which has a domain in _L2(Ω)$L_2(\Omega )$ that does not lie within the usual Sobolev Hilbert function space W²(Ω)$W^2(\Omega )$. These _THar$T_{\mathrm{Har}}$-like operators cannot be specified by conventional differential boundary conditions on the boundary of ∂Ω$\mathrm{\partial }\Omega$, and may have non-empty essential spectra.     

A surjectivity theorem for differential operators on spaces of regular functions

In this article we show that it is possible to construct a Koszul-type complex for maps given by suitable pairwise commuting matrices of polynomials. This result has applications to surjectivity theorems for constant coefficients differential operators of finite and infinite order. In particular, we construct a large class of constant coefficients differential operators which are surjective on the space of regular (or monogenic) functions on open convex sets.