This is a short survey on the connection between general extension theories
and the study of realizations of elliptic operators A on smooth domains in R^n,
n > 1. The theory of pseudodifferential boundary problems has turned out to be
very useful here, not only as a formulational framework, but also for the
solution of specific questions. We recall some elements of that theory, and
show its application in several cases (including recent results), namely to the
lower boundedness question, and the question of spectral asymptotics for
differences between resolvents.Comment: 26 pages, style changed to LaTeX, new material added at the end, to
appear in the Lecture Notes Series of the London Math. Soc. published by
Cambridge Univ. Pres
