Positivity 8: 243–256, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Positive Invertibility ofNonselfadjoint Operators
Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105,
Israel. E-mail: email@example.com
(Received 1 November 2001; accepted 15 July, 2003)
Abstract. The paper deals with a class ofnonselfadjoint operators in a separable Hilbert lattice. Con-
ditions for the positive invertibility are derived. Moreover, upper and lower estimates for the inverse
operator are established. In addition, bounds for the positive spectrum are suggested. Applications to
integral operators, integro-differential operators and inﬁnite matrices are discussed.
AMS (MOS) Subject Classiﬁcation: 47A55, 47A75, 47G10, 47G20
Key words: linear operators, positive invertibility, spectrum, integral operators, integro-differential
operators, inﬁnite matrices
1 Introduction and statement ofthe main result
Let H be a separable Hilbert lattice [9, p. 128] with a scalar product , a norm
and the unit operator I. For a linear operator A, Ais the spectrum, DomA
is the domain.
In the present paper we consider a class oflinear operators in H. New conditions
for the positive invertibility are derived. They supplements the well-known results,
cf. [8, 10] and references therein.
A linear operator V is a Volterra one ifit is quasinilpotent (that is, V= 0)
and compact, cf. the book . Recall that a maximal resolution of the identity
Pt − t is a left continuous orthogonal resolution of the identity,
such that any gap Pt
of Pt (ifit exists) is one-dimensional, cf. [3,
p. 69]. Similar notions can be found in [2, 6].
A maximal resolution of the identity P is said to be positive, if the operators
are nonnegative in the sense ofthe order.
In this paper we investigate an operator ofthe form