ACTA issues

Decomposition of selfadjoint projections in Krein spaces

Alejandra Maestripieri, Francisco Martínez Pería

Acta Sci. Math. (Szeged) 72:3-4(2006), 611-638

Abstract. Given a Hilbert space $({\cal H}, \langle, \rangle )$ and a bounded selfadjoint operator $B$ consider the sesquilinear form over ${\cal H}$ induced by $B$, $$\langle x,y\rangle_B=\langle Bx,y\rangle, x,y\in{\cal H}. $$ A bounded operator $T$ is $B$-selfadjoint if it is selfadjoint with respect to this sesquilinear form. We study the set ${\cal P}(B,{\cal S})$ of $B$-selfadjoint projections with range ${\cal S}$, where ${\cal S}$ is a closed subspace of ${\cal H}$. We state several conditions which characterize the existence of $B$-selfadjoint projections with a given range; among them certain decompositions of ${\cal H}$, $R(|B|)$ and $R(|B|^{1/2})$. We also show that every $B$-selfadjoint projection can be factorized as the product of a $B$-contractive, a $B$-expansive and a $B$-isometric projection. Finally two different formulas for $B$-selfadjoint projections are given.

AMS Subject Classification (1991): 47A07, 46C20, 46C50

Keyword(s): Indefinite metric, Krein space, oblique projections, selfadjoint operators

Received November 16, 2005, and in revised form July 10, 2006. (Registered under 5940/2009.)