Positivity 1: 239–254, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Connectivity of Level Sets of Quadratic Forms and
Hausdorff-Toeplitz Type Theorems
and ALEXANDER MARKUS
Department of Mathematics, Technion, Haifa 32000, Israel.
Department of Mathematics and Computer Sciences, Ben-Gurion University of the Negev,
Beer-Sheva 84105, Israel.
(Received: 7 February 1997; accepted in revised form: 21 July 1997)
Abstract. Main theorem: for an arbitrary linear operator
X ! X
in a complex pre-Hilbert
3, all level sets
fx 2 X
hAx; xi = ; kxk =
are connected. This fails if
is the numerical range. The main theorem implies the
known result on convexity of generalized numerical range of three Hermitian operators.
Mathematics Subject Classiﬁcation (1991): 15A60, 15A63.
Keywords: quadratic forms,connectivity, convexity,generalizednumerical range, Hausdorff-Toeplitz
be a complexlinear space provided with a positive deﬁnite innerproduct
is a pre-Hilbert space). Let
X ! X
be a linear operator. The numerical
is deﬁned as the set of values of the quadratic form
S = fx
x 2 X; kxk =
g; kxk =
By the classical Hausdorff-Toeplitz theorem ,  this set is convex. In his
original proof Hausdorff observed that for any Hermitian operator
the level sets
x 2 S; hHx; xi = g
are connected. Then for any pair of Hermitian operators
on an above mentioned level set form a connected
subset of the real axis, i.e. an interval of R. This immediately leads to the required
A = H
is the angle in the complex plane C
between R and the segment
corresponding to a given pair
At present a lot of proofs of the Hausdorff-Toeplitz theorem are known. Let
us only mention that the theorem follows from its 2-dimensional case where the
is either an elliptic disk or a segment, or a single point  (see also [4,