Positive and negative spectral projections are maps of class $C^{\infty}$
Keywords:
Spectral theory, complexification of linear operators, negative and positive, spectral subspaces, orthogonal projection, Cauchy’s integral formula.Abstract
Let H be a real or complex Hilbert space. We denote by GlS(H) the set consisting of self-adjoint bounded isomorphism. If L 2 GlS(H), then there exist a L-invariant splitting H = H+(L) H?(L); such that L is positive on H+(L) and negative on H?(L). The main goal of this work is to give an elementary prove of that P?; P+ : GlS(H) ! LS(H), where P?(L) and P+(L) are the orthogonal projections onto H?(L) and H+(L) respectively, can be expressed as
P?(L) = ? 1 2i Z ? (L ? I)?1d and P+(L) = I + 1 2i Z ? (L ? I)?1d;
where ? is a closed path containing the negative spectrum of L in its interior. Using this representation, we will
see that P? and P+ are C1-maps.
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