Editorial Preface
During the second half Of the 20th century, Operator theory has been developed from a subdomain Of functional analysis tO a fast increasing domain by itself. The spec-tral theory Of linear Operators has been recognized from the early beginning as the most important, mainly due tO its applications tO ordinary and partial differential equations. The monumental three volume monograph Of Nelson Dunford and Jacob T. Schwartz published under the title Linear Operators was meant tO be a panorama Of the domain and an ultimate source Of reference, as well. In this respect, the class Of the so—called spectral operators in the Dunford sense was considered for some time as the peak Of the spectral theory Of linear operators. However, it was soon admitted that beyond this class there existed a vast field Of investigations. The theory Of spectral operators, developed by Dunford and his collaborators, was based on spectral measures, while the new direction Of investigation was based on spectral subspaces or spectral functional calculi.
A major breakthrough was the introduction Of the notions Of decomposability, spec-tral distribution and maximal specspec-tral space by C. Foias [Bul1. Sci. Math. (2) 84 (1960), 147—158]. In 1962, Ion COloara published the paper Generalized Spectral Operators [Rev. Roum. Math. Pures Appl. 7 (1962), 459—465] where he introduced the class Of generalized spectral Operators, that is, bounded linear operators T on a Banach space X such that (i) T is decomposable; (ii) there exists a spectral distribution U such that T commutes with U (i.e., TUf = UfT for all f); (iii) T and U), have the same maximal Spectral spaces, where /\ is tO be interpreted as the mapping /\ ~> A. TO have an idea Of what this definition is gOOd for, let us mention that the author proved that an operator T is a generalized spectral Operator if and only if T : S + Q, where S is a generalized scalar Operator in the sense Of Foias, and Q is a quasinilpotent Operator which commutes with U. In addition, if T is a generalized spectral Operator, then there exist two Operators R and J such that (1) T = R + U and RJ 2 JR, (2) R and J have real Spectra, (3) R is a generalized scalar Operator and J is a generalized spectral Operator, and (4) if R1 and J1 satisfy (1) and (2) and commute with R, then there exists a quasinilpotent Operator Q1 such that R1 = R + Q1 and J1 : J +7121.
It should be mentioned that independently, Fumy-Yuki Maeda gave in [Pacific J. Math. 13 (1963), 177—192] a definition Of generalized spectral Operators in locally convex spaces and it was Ion COloara who proved one year later in [Stud. Cerc. Mat. 15 (1964), 499—536] that for the case Of Banach spaces the two notions coincide.
These results turned out to be Of a crucial importance in the spectral theory Of linear Operators and soon, many other mathematicians have considered applications and gener-alizations. Some Of the early results have been gathered in the celebrated monograph Of Ion COloara and Ciprian Foias Theory of Generalized Spectrai Operators [Mathematics
xii Editorial Preface
and its Applications, Vol. 9. Gordon and Breach, Science Publishers, New York-London— Paris, 1968, xvi+232 pp.] that gave a systematic account of this research.
After forty years from these mathematical events, we invited a few mathematicians working in this field to present an overview of the deveIOpments that have occurred in the meantime and to celebrate the 70th birthday of one of the main contributors to the devel— opment of spectral theory in Banach spaces, Ion Colojoara. This was organized originally as a one day conference in the Institute of Mathematics of the Romanian Academy. It came as natural to persuade the participants as well as other mathematicians that were unable to attend this meeting, to write down their presentations and to collect all these in a volume dedicated to the 70th anniversary of Ion Colojoara. We would like to thank all the contributors for the enthusiasm they showed immediately and for the time and energy they spent on this project.
The volume consists of invited survey papers of mathematicians that have con— tributed to the development of the ideas promoted by Ion Colojoara in spectral theory and its applications, as well as a careful selection of a short number of research papers of some of his collaborators, former students and prominent mathematicians working in the field. The topics include: growth properties of the resolvent of bounded linear operators on a Banach space near a thin part of their spectra, and their relation to hyperinvariant subspaces, functional calculus properties, spectral decomposition properties, and pertur— bations by compact operators; Operator-valued positive measures, in particular spectral measures, viewed as mathematical objects related to spectral decompositions of linear operators, and their applications to moment problems; special properties of Kolmogorov decompositions of positive definite operator-valued functions; realizations of norm un— bounded rings of projections by means of an unbounded, closed, densely defined operator and their applications to unbounded spectral measures; invariance of the maximal spec-tral spaces of decomposable operators and the invariance of Dunford specspec-tral spaces of general bounded operators on a complex Banach space; Hilbert space realization tech— niques for analytic functions in the unit ball and unit polydisc in C” and their applications to submanifolds of the unit polydisc and of the unit ball which possess a strict bounded analytic extension property; perturbation theory of selfadjoint Operators in Krein spaces and local variants of definitizability; spectral resolutions for self—adjoint and normal 0p-erators in Krein spaces having invariant maximal non—negative pseudo—regular subspaces; classes of orthogonal polynomials in several non—commuting variables with applications
to a non—commutative version of the orthogonal polynomials on the real line; idempotent
linear relations and generalized projections on a Hilbert space with some applications to spectral theory.
Finally, we would like to express our gratitude to Daniel Beltit‘a for converting most
of the files to a unifying BTEXZE style and to Cristina Gheondea for technical editing and
proofreading the whole volume. Funds for editing, printing and distributing this volume have been provided by the Editorial Office of the Journal of Operator Theory.
