Eigenfunction Expansions for Nondensely De- Fined Operators Generated by Symmetric Ordinary Differential Expressions by Earl A. Coddington

where the pk are complex-valued functions of class C k on an interval a < x < b, and pn(x) / 0 there. In the Hilbert space § = 2 {a, b) let S0 be the closure in § 2 of the set of all {ƒ, Lf} for ƒ e C$(a, b), the functions in C°°(a,b) vanishing outside compact subintervals of a < x < b. This S0 in a closed densely defined symmetric operator whose adjoint has the domain 3>(S$), the set of all ƒ e C~ \a,b) such that f { n ~ 1 } is absolutely continuous on each compact subinterval and Lfe9). For feT)(S$), S%f=Lf. If M 0 = S* 0 S0, then dimlMo)* = dim D P o ) 1 ) = dim v(S* + il) = co9 say (v(T) = null space of T). Thus 0 ^ co :g n, and dim M 0 = co + + co~ ^ 2n. Let £>0 be a subspace of § , dim Jr>0 = P < °°> a n d define the operator S, with D(S) = D(S0) n ( § 0 §0), via S c S0. We see that (2.1) of [2] is satisfied and Theorem 1 of [2] is applicable to S. If co = co~ = co, which we now assume, then Theorem 2 of [2] is also applicable. For u,v e D(S*) we have Green's formula