On the Spectral Decomposition of Dichotomous and Bisectorial Operators

TL;DR

This paper investigates the spectral properties of unbounded operators on Banach spaces, establishing invariant subspaces related to their spectra and providing explicit formulas for associated projections, with applications to bisectorial operators.

Contribution

It proves the existence of invariant subspaces for such operators under resolvent boundedness and derives explicit formulas for projections, extending to bisectorial operators.

Findings

Invariant subspaces exist under resolvent boundedness.

Explicit formulas for projections via resolvent integrals.

Results apply to bisectorial operators with simplifications.

Abstract

For an unbounded operator $S$ on a Banach space the existence of invariant subspaces corresponding to its spectrum in the left and right half-plane is proved. The general assumption on $S$ is the uniform boundedness of the resolvent along the imaginary axis. The projections associated with the invariant subspaces are bounded if $S$ is strictly dichotomous, but may be unbounded in general. Explicit formulas for these projections in terms of resolvent integrals are derived and used to obtain perturbation theorems for dichotomy. All results apply, with certain simplifications, to bisectorial operators.