Dense linear manifolds and application to linear differential operators

Abstract

In an operator theory of perturbed ordinary differential operators, it is important to know whether the perturbations are closed or densely defined. All known general theorems for checking whether they are closed seem to require that the perturbing term is T-compact or have a restriction on its Tnorm (see, for example, Theorems IV.I.1, IV.I.11 of [5], Lemma V.3.5 of 13 ], and Proposition 2.5 of [8)). However, these theorems are ineffective or inconvenient to apply to the operator generated by countably many ordinary differential expressions. Thus, the main purpose of this paper is twofold: First we give new characterizations (Theorems 1,2 below) which are useful for testing the closedness of a perturbed linear operator and do not require any argument on compactness or T-norm. Secondly, using these characterizations we show in Theorems 3,4 below that a large class of operators subject to infinitely many abstract boundary conditions have densely defined domains. The main tool used here is an abstract adjoint theory. We now fix some notations. If X is a Banach space, then x* will denote the Banach space of all continuous conjugate linear functionals on X. If M is a linear manifold, then, M*, *M, MC, ‘M will denote the adjoint, the preadjoint, the closure, and the w*-closure of M, respectively. If q is a linear operator whose graph is contained in the direct sum X, @X, of Banach spaces X, and X,, then Y’f will denote the (possibly) multi-valued operator whose graph is (graph Y,)*. A similar definition applies to *p: if Yl is a linear operator whose graph is contained in g @ fl. For definitions of adjoint and preadjoint, see [2] or [9]. If D, and D, are m x it and q x r matrices, then D, @D, will denote the (m + q) x (n + r) block matrix by joining the lower right corner of D, to the upper left corner of D,. The algebraic sum of M, and M, is denoted by M, -iM,. The Hilbert space of all 1 x N complex constant matrices a with aa* < 00 (a * the conjugate transpose of a) is