Physiopathologie du retrécissement valvulaire aortique: Progrès, incertitudes, implications thérapeutiques

Abstract

This chapter discusses the generalizations of self-adjointness to Banach spaces. The theory of self-adjoint operators in Hilbert spaces is the best-developed and most important in the spectral theory of linear operators. Because of the elegance and importance of this theory, there have been many attempts to extend some of these results to classes of linear operators in complex Banach spaces. The operators discussed in the chapter are seen to have a resolution of the identity in a suitably generalized sense with at most a mild spectral singularity at zero. The chapter presents an overview of Dunford's spectral operators, Lumer's Hermitian operators, and Stampfli's adjoint abelian operators. Because a projection operator need not be Hermitian in the original norm of a Banach space, a positive scalar operator need not be Hermitian with respect to the given norm.