Comparison of the states of closed linear transformations

John Douglas Faires

Vol. 54 (1974), No. 1, 123–133

DOI: 10.2140/pjm.1974.54.123

Abstract

Let X and Y be Banach spaces and T, respectively S, be a bounded linear transformation mapping X into Y , respectively Y into X. It is well-known that a nonzero complex number λ belongs to the spectrum of ST precisely when λ belongs to the spectrum of TS. The main result of §2 shows that for λ≠0 the states of the operators ST − λI_X,TS − λI_Y agree.

Sufficient conditions are obtained for this same result to hold when T and S are unbounded closed linear transformations from X into Y and Y into X respectively. Section 4 compares spectral decompositions of ST and TS when these sufficient conditions are satisfied.

Mathematical Subject Classification 2000

Primary: 47A10

Milestones

Received: 16 March 1973

Revised: 21 September 1973

Published: 1 September 1974