Generators of positive C₀-semigroups

Shizuo Miyajima and Noboru Okazawa

Vol. 125 (1986), No. 1, 161–176

DOI: 10.2140/pjm.1986.125.161

Abstract

By the use of abstract Kato’s inequality for generators of positive C₀-semigroups, it is shown that a differential operator with smooth coefficients (with natural domain) is of order at most 2 and (degenerate) elliptic provided it generates a positive C₀-semigroup on L^p(Rⁿ) (1 ≤ p < ∞). Conversely it is also shown that a certain second order elliptic differential operator with a singular 0th order coefficient generates a positive C₀-semigroup on L^p(Rⁿ) (1 < p < ∞).

Mathematical Subject Classification 2000

Primary: 47D05, 47D05

Secondary: 35G05, 47B38, 47F05

Milestones

Received: 2 October 1984

Revised: 26 December 1985

Published: 1 November 1986