Graduate Courses

Singular Integral Operators  1994/1996

Linear Operators  1996/1997

Algebras of Singular Integral Operators  1995/1996, 1997/1998,
2003/2004
Program:
Fredholm
operators and the Calkin algebra
Characterization of Fredhom operators. Perturbation of Fredholm
operators and index properties. SemiFredholm operators and their
generalized inverse.
The Cauchy singular integral operator
Classical perspective and the Cauchy singular integral operator in
Lebesgue spaces. Generalizations to weighted Lebesgue spaces and to
systems of curves. The transpose of the Cauchy singular integral
operator.
Singular intgeral operators and the
factorization theory
The concept of factorization of functions. The singular integral
operator with rational coefficients. The singular integral operator
with continuous coefficients. Factorization in decomposing Banach
algebras. Generalized factorization of functions. Some results
concerning the factorization of functions with a finite number of
discontinuities. Generalization to the matrix case of the results about
factorization of functions. Application to boundary valued problems
from diffraction theory.
The Gelfand theory and generalizations to
noncommutative algebras
The Gelfand transformation and some of its applications. Ideals and
invertibility. Localization principles. PIalgebras. Introduction to
the local trajectory method.
Invertibility in algebras of singular
integral operators and in Toeplitz algebras
Application of the twoidempotent theorem to the algebra generated by
the Cauchy singular integral operator and by multiplication operators
by piecewise continuous functions on the unit circle. The algebra
generated by Toeplitz plus Hankel operators. Localization over orbits
and the invertibility in C*algebras with shift operators.
Program:

Operator Algebras  2000/2001, 2001/2002, 2002/2003, 2004/2005
General
results for Banach algebras:
Spectrum and spectral radius. The Gelfand representation theory for
commutative Banach algebras. Functional calculus. Spectral theorem.
C* Algebras:
The Gelfand Naimark theorem for commutative algebras. The functional
calculus for normal operators.
Positive elements. Positive linear functionals. States and pure states.
The GelfandNaimarkSegal construction. The GelfandNaimark theorem.
Liminal and postliminal C* algebras. Von Neumann algebras. Double
commutant theorem. Kaplansky density theorem. Crossed products. The
isomorphism theorem for commutative groups and algebras.
Localtrajectory method.
Representations of Banach algebras:
Primitive ideals. Primitive algebras. Irreducible representations. The
Jacobson radical. The Jacobson topology.
Representations of algebras that satisfy a polynomial identity.
Algebras generated by two idempotents.
The AllanDouglas localization principle and the representation theory.
Undergraduate Courses
• Applied Mathematics
for Electrical Engineers I, 19781983.
• Applied
Mathematics
for Electrical Engineers II, 19781983.
• Linear
Algebra, 19981991.
• Mathematical
Analysis I
for students
of:
 Civil
Engineering; Territorial Engineering, 199293 .
 Mechanical
Engineering; Mining and Geological Engineering; Naval Architecture and
Marine
Engineering, 199394.
 Information
Systems and Computer Engineering, 199495.
 Chemical
Engineering; Mining and Geological Engineering, Naval Architecture and
Marine
Engineering, Materials Engineering, 199596.
 Information
Systems and Computer Engineering, 199798.
 Mechanical
Engineering, Mining and Geological Engineering, Naval Architecture and
Marine
Engineering, Materials Engineering, 19992000.
 Mechanical
Engineering, Mining and Geological Engineering, Naval Architecture and
Marine
Engineering, Materials Engineering, 20002001.
 Mechanical
Engineering, 20012002.
 Civil
Engineering, Territorial Engineering,
Naval Architecture and Marine Engineering, 20042005.
• Mathematical
Analysis II
for
students
of:
 Civil
Engineering; Territorial Engineering, 199293.
 Mechanical
Engineering; Mining and Geological Engineering; Naval Architecture and
Marine
Engineering, 199394.
 Information
Systems and Computer Engineering, 199495.
 Chemical
Engineering; Mining and Geological
Engineering, Naval Architecture and Marine
Engineering,
199596.
 Civil
Engineering, Territorial Engineering, 19992000.
• Differential
and Integral Calculus I

for students of
Information Systems and Computer Engineering,
20062007.
Lectures
Notes
• Bastos, M. A.,
Santos,
P.A, “Introduction
to Operator Algebras” Textbook in progress.
• Bastos, M. A., Lebre, A. B.:
"Apontamentos de Álgebras de Operadores Integrais Singulares", 280pp, 1998.
• Bastos, M. A.: "Noções de
Álgebra,
Trigonometria e Análise Matemática "Curso Internacional
de Hidrologia
Operativa, Manual, Vol. I, 137, 1984.
