The
main objective of the
project is to develop the invertibility theory for classes of non-local
C*-algebras which allows one to consider mathematical models of
processes in
systems where the interactions between its parts are delayed. The
influence of
delays is essential in different fields namely in control theory,
immunology
and epidemiology. The main operators investigated in this project are
natural
generators of non-local algebras and the research pursues in two
directions: a
research line in Operator Algebras and a research line that lies at the
intersection of Operator Theory and Complex Analysis.
Crossed product C* algebras encode all the information of the original
C*
algebras and group actions on these algebras. A method of localization
on
orbits, the local trajectory method developed by Antonevich, Lebedev,
[AL] and
Karlovich, [K], allows one to study invertibility in some non local
classes of
these algebras. Recently this tool was applied to classes of algebras
of
operators with oscillatory coefficients, [BKS]. The theory has
undergone new
developments in recent years and progress on the localization on orbits
is
expected. One of the goals of the project is the construction of
spectral
measures and the generalization of the local trajectory method for
C*-algebras
of operators on Hardy and Bergman spaces.
Recognizing similar features on the problem of applicability of
approximation methods in very different contexts Roch and Silbermann
defined
the standard model algebra, [HRS], as an abstract algebra of
approximate
sequences of operators on a Hilbert space satisfying a set of axioms.
These
algebras open a new way to study linear bounded operators. One of the
goals of the
project is the development of the standard model algebra theory by
studying new
classes of approximate sequences of operators in non-local type
algebras, and
by determining the defect numbers of classes of linear bounded
operators.
Several problems of the theory
of linear operators appear naturally connected with the investigation
of
operator algebras. Another research line is the Operator Corona
Problem. The
Operator Corona Problem corresponds to finding necessary and sufficient
conditions for F in H(X,Y) to have left inverse. We intend to
contribute to
this area and get invertibility results for classes of linear bounded
operators
using the new developments introduced by N. Nikolski in [N]. One of the
goals
of the project is to explore the Operator Corona Theorem in order to
get
constructive methods for the invertibility of Toeplitz-Hankel
operators. The research lines will be developed in connection with international
research
groups from Mexico, Germany and France. The connection with the French
group
will be done for the first time with this project.
[AL] A.
Antonevich and A.
Lebedev, Functional Differential Equations: I. C*-Theory- Pitman
Monographs
and Surveys in Pure Applied Mathematics Vol. 70, Logman Scientific
&
Technical, 1994.
[BKS] A. Böttcher, Yu.
Karlovich and B. Silbermann, Singular Integral Equations with PQC
Coefficients
and Freely Transformed Argument, Math. Nach. 166, 1994, 113-133.
[HRS] R. Hagen;
S. Roch; B. Silbermann, C*-algebras and numerical analysis. Monographs and Textbooks in Pure
and
Applied Mathematics, 236. Marcel Dekker, Inc., New York,
2001. 376 pp.
ISBN: 0-8247-0460-6 (Reviewer: Georg Heinig).
[K] I. Yu. Karlovich, The
Local-Trajectory Method of Studying Invertibility in C*- Algebras of Operators With Discrete Groups of Shifts, Soviet
Math. Dolk.
Vol.37, nº 2, 1988.
[N]
N. Nikolski, Operators, Functions, and Systems: An Easy
Reading. Americal
Mathematical Society 2002.