ACTA issues

Oblique projections and Schur complements

Gustavo Corach, Alejandra Maestripieri, Demetrio Stojanoff

Acta Sci. Math. (Szeged) 67:1-2(2001), 337-356

Abstract. Let ${\cal H}$ be a Hilbert space, $L({\cal H} )$ the algebra of all bounded linear operators on ${\cal H}$ and $ \langle, \rangle_A \colon{\cal H} \times{\cal H} \to{\msbm C}$ the bounded sesquilinear form induced by a selfadjoint $A\in L({\cal H} ) $, $ \langle\xi, \eta\rangle _A = \langle A \xi, \eta\rangle, \xi, \eta\in {\cal H}.$ Given $T\in L({\cal H} )$, $T$ is $A$-selfadjoint if $AT = T^*A$. If ${\cal S} \subseteq{\cal H}$ is a closed subspace, we study the set of $A$-selfadjoint projections onto ${\cal S} $, $$ {\cal P}(A, {\cal S} ) = \{Q \in L({\cal H}) : Q^2 = Q, R(Q) = {\cal S}, AQ = Q^*A\} $$ for different choices of $A$, mainly under the hypothesis that $A\ge0$. There is a closed relationship between the $A$-selfadjoint projections onto ${\cal S} $ and the shorted operator (also called Schur complement) of $A$ to ${\cal S} ^\perp $. Using this relation we find several conditions which are equivalent to the fact that ${\cal P}(A, {\cal S} ) \not =\emptyset $, in particular in the case of $A\ge0$ with $A$ injective or with $R(A)$ closed. If $A$ is itself a projection, we relate the set ${\cal P}(A, {\cal S} ) $ with the existence of a projection with fixed kernel and range and we determine its norm.

AMS Subject Classification (1991): 47A64, 47A07, 46C99

Received May 2, 2000, and in revised form September 27, 2000. (Registered under 2787/2009.)