Notes on generalized boundary value problems in Banach spaces. I. Adjoint and extension theory

Richard Clark Brown

Vol. 85 (1979), No. 2, 295–322

DOI: 10.2140/pjm.1979.85.295

Abstract

Let A : X → Y be a densely defined closed operator where X and Y are Banach spaces. Let F be a locally convex topological vector space and H : X → F an operator such that N(H) and D(A) have nontrivial intersection and D(H^∗) is total over F. We compute A_H^∗ and A_H^∗ where A_H is the operator determined by A on N(H) and A_H(x) = (Ax,Hx)^t.

We also characterize certain closed extensions of A_H and the adjoints of these extensions. In particular application is made to the problem of determining self-adjoint extensions of symmetric operators restricted by boundary conditions in a Hilbert space.

Mathematical Subject Classification 2000

Primary: 47A05

Secondary: 34G10

Milestones

Received: 31 July 1978

Published: 1 December 1979