Stieltjes differential-boundary operators. III. Multivalued operators–linear relations
Allan M. Krall · Pacific Journal of Mathematics · 1975
This article deals with a multivalued differential-boundary operator on a nondense domain regarding it as a linear relation.The adjoint relation is derived.It is shown that these dual relations have the same form as exhibited in earlier papers where the operators involved were uniquely defined on dense domains.Self-adjoint relations are considered on the Hubert space ^2[0,1].The connection with self-adjoint operators defined on subspaces of -£^[0,1] is made.L Introduction* This article is a continuation of [8] and [9].The notation is the same.We review it briefly.X is the Banach space ^f%[0, 1], 1 w ^ere ^ an( i -B are m χ/ ^ matrices satisfying rank (A: 5) = m, and C and 5 are (2n -m x ri) matrices.Hence the large matrices above may be multiplied together in the usual component-like manner.K is a regular m x n matrix valued function of bounded variation satisfying dK(0) = 0, dK(l) = 0. iξ is a regular r x n matrix valued function of bounded variation satisfying dK^O) -0, dK^ΐ) = 0.H is a regular w x (2m -m) matrix valued function of bounded variation satisfying dH(0) = 0, ώJΪ(l) = 0. jff x is a regular n x s matrix valued function of bounded variation satisfying dH^O) = 0, dffi(l) = 0. P is a continuous ^ x ^ matrix.Now let 3f denote those elements yeX satisfying