Analog of the Riesz theorem and the basis property in L p of a system of root functions of a differential operator: I

Abstract

An ordinary differential operator of arbitrary order is considered. We find necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of a system of root functions of this operator in L p . V.A. Il'in is known to be the first to pose the problem as to whether the Riesz inequality [see formula (2) below] holds for a function system that is not complete or orthonormal. He proved the inequality for systems of root functions of the Laplace and Schrödinger operators [1]. The Bessel property and the unconditional basis property were studied for systems of root functions of ordinary differential operators of order ≥ 2 in L 2 in [2-10]. Necessary conditions for the Riesz inequality to hold for a system of root functions of a differential operator of arbitrary order were established in [11]. In the present paper, we prove an analog of the Riesz theorem (see [12, p. 154]) and the theorem on the basis property in L p for systems of root functions of a differential operator.Note that the results of the present paper were announced in [13,14]. Here we present a detailed proof. BASIC NOTIONS AND STATEMENT OF RESULTSOn the interval G = (0, 1), consider a differential operator Lu = u (n) + P 1 (x)u (n−1) + P 2 (x)u (n−2) + · · · + P n (x)u,where P (x) ∈ L 1 (G), = 1, . . . , n. By D we denote the class of functions absolutely continuous together with their derivatives of order ≤ n − 1 on the closed interval G.Following Il'in, we understand the root functions of L in the generalized sense, admitting completely arbitrary boundary conditions [2]. Consider an arbitrary system {u k (x)} of root functions of the operator (1). Let {λ k } be the corresponding system of eigenvalues of this operator. We require that, together with each associated function of order s ≥ 1, the system {u k (x)} also contains the corresponding eigenfunction and all associated functions of order < s. This means (see [15]) that each element u k (x) of the system {u k (x)} other than identical zero belongs to the class D and almost everywhere in G satisfies either the equation Lu k + λ k u k = 0 [in this case, u k (x) is an eigenfunction, and λ k is an eigenvalue] or the equation Lu k + λ k u k = u ν(k) , where the index ν(k) is uniquely determined by the index k and ν(k 1 ) = ν(k 2 ) for k 1 = k 2 . [In the latter case, λ k = λ ν(k) , u k (x) is an associated function of order r ≥ 1, and u ν(k) is an associated function of order r − 1.] 7

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

“…We have to show that the resulting formal solution in the form of series (17), (18) satisfies equation (4) and conditions ( 5), (7). Let us first show that series (17), (18), as well as the formal derivative with respect to the variable t and formal derivatives up to the fourth order with respect to In [14] the validity of the estimates max X…”

“…In [9], the authors of this work proposed a new approach to prove the uniform convergence of formally differentiated series, which represent a formal solution to the inverse problem for the equation of a fourth-order hyperbolic equation with complex-valued coefficients. This approach has two advantages: 1) the first advantage is the use of estimates of the norms of eigenfunctions derivatives through the norm of eigenfunctions [14]; the second advantage is the use of the properties of uniform boundedness of Riesz bases consisting of eigenfunctions of the differential operator [15]. In this section, this approach is developed for the case of inverse problems for a fourth-order parabolic equation with complex-valued coefficients.…”

Properties in L p of root functions for a nonlocal problem with involution

“…Applications of the spectral approach for PDEs with involution and/or nonlocal boundary conditions are discussed in [1,2,25,27,[29][30][31]. For the spectral properties of conventional differential operators in non-Hilbert spaces, one could refer to [4,6,7,19,20,34].…”