Positivity 2: 153–164, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
The Theory of Operators with Dominant Main
YURY YA. AGRANOVICH
Area Centre for New Information Technologies, Department of the Control in a Social Realm and
Medicine, Voronezh State Technical University, Voronezh, Russia
(Received: 17 December 1996; accepted in revised form: 13 February 1998)
Abstract. In this paper a characterization of the symmetric operators on a ﬁnite dimensional Hilbert
space which have a matrix representation with a dominant diagonal with respect to any orthonormal
basis are obtained. The set of such operators is a solid, reproducing, normal and acute cone in
the space of symmetric operators. These results are applied to localizing the spectrum of operators
Mathematics Subject Classiﬁcations (1991): 15A12, 15A42, 15A48, 15A22
Key words: matrices with dominant main diagonal
It is well known that matrices with dominant main diagonal are of great importance
for numerical methods. However, studying localization problems of eigenvalues of
polynomial operator pencil, investigating the behavior of the spectrum of operator
functions and some other fundamental problems, it is desirable, on the one hand,
to make use of the convenient properties of matrices with dominant main diagonal,
and on the other hand it is necessary to obtain results in operator terms which do
not depend of the choice of the orthonormal basis. Therefore it is natural to put the
• do there exist operators acting in space of ﬁnite dimension whose matrices
have a dominant main diagonal in any orthonormal basis ?
There is a simple answer: such operators exist, for example, operators of the form
λI. But indeed, we shall show in the following, that the answer is not exhausted by
these trivial operators.
This paper is devoted to investigation of some properties of the operators which
answer the above question. We shall also give some examples of their applications.
Consider the space S of symmetric operators which act in the n-dimensional
real Hilbert space H. We shall denote by D
the set of operators A, whose matrices
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