A spectral duality theorem for closed operators

I. Erdélyi and Sheng-Wang Wang

Vol. 114 (1984), No. 1, 73–93

DOI: 10.2140/pjm.1984.114.73

Abstract

The spectral duality theorem asserts that a densely defined closed operator T induces a spectral decomposition of the underlying Banach space X iff the conjugate T^∗ induces the same type of spectral decomposition of the dual space X^∗. This theorem is known for bounded linear operators in terms of residual (S)-decomposability. In this paper we extend the spectral duality theorem to unbounded operators, under a general type of spectral decomposition. Our approach to the spectral duality leads through the successive conjugates T^∗, T^∗∗ and T^∗∗∗ of T, under their domain-density assumptions.

Mathematical Subject Classification 2000

Primary: 47B40

Secondary: 47A10

Milestones

Received: 24 September 1982

Revised: 14 September 1983

Published: 1 September 1984