Positivity 13 (2009), 759–770
2009 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040759-12, published online February 6, 2009
Congruence of selfadjoint operators
Guillermina Fongi and Alejandra Maestripieri
Abstract. Given a bounded selfadjoint operator a in a Hilbert space H,the
aim of this paper is to study the orbit of a, i.e., the set of operators which are
congruent to a. We establish some necessary and sufﬁcient conditions for an
operator to be in the orbit of a. Also, the orbit of a selfadjoint operator with
closed range is provided with a structure of differential manifold.
Mathematics Subject Classiﬁcation (2000). Primary 47B15, 58B20.
Keywords. Selfadjoint operators, congruence of operators, differential geometry.
Let L(H) be the algebra of linear bounded operators of a separable Hilbert space H
and GL(H) the group of invertible operators of L(H). Two operators a, b ∈ L(H)
are congruent if there exists g ∈ GL(H) such that b = gag
.Ifa and b are positive,
then a and b are congruent if and only if their ranges R(a)andR(b) are unitarily
equivalent, i.e. there exists a unitary operator u such that R(b)=uR(a), see .
In , it was proved that if a and b are invertible selfadjoint operators, b is con-
gruent to a if and only if the reﬂections of their polar decompositions belong to
the same unitary orbit. The purpose of this paper is to study the set of operators
which are congruent with a, when a is a given selfadjoint operator, not necessarily
The congruence between selfadjoint operators deﬁnes a natural action of the
group GL(H) over the set L(H)
of selfadjoint operators given by L
a = gag
the set L(H)
is the (disjoint) union of the orbits of this action. In , Corach
et al. described the set GL(H)
, of selfadjoint invertible operators, as a reductive
homogeneous space of GL(H), with a canonical connection and a Finsler metric.
They proved that any pair of points a, b ∈ GL(H)
having the same unitary part
in the polar decomposition, can be joined by a geodesic. These geodesics are short
if measured with the (Finsler) metric. Observe that GL(H)
is the union of the
orbits of invertible selfadjoint operators. In particular, the orbit of the identity is
, the cone of positive invertible operators.
The study of the geometric structure of the set of selfadjoint operators, in
the non-invertible case, was continued in , ,  and . The set L(H)