Functional Analysis and Applications Seminar 

03/05/2002, 14:00 — 15:00 — Room P3.10, Mathematics Building
Maria Amélia Bastos, Instituto Superior Técnico, U.T.L.

Fredholm Theory in an Algebra Generated by OperatorswithOscillating Symbols and the Cauchy Singular Operator

In this talk we deal with a Fredholm theory in the algebra $A$ generated by operators with oscillating symbols and the Cauchy singular operator as a subalgebra of bounded linear operators in ($n$-vector) Lebesgue spaces.

In the first part of the talk, using the limit operator theory, a necessary Fredholmness condition for any operator in $A$ is established and, from one of the local principles, sufficient Fredholmness conditions for some elements in this algebra are obtained. A Fredholm criterion for Toeplitz operators with matrix symbols in the $C^{*}$ algebra $C$ generated by slowly oscillating and semi-almost periodic matrix functions on $ℝ$ is established as a consequence.

In the second part, using the notion of harmonic extension, an index theory for Fredholm Toeplitz operators whose generating functions belong to the $C^{*}$ algebra $C$ is developed. The most relevant result is obtained by reducing to Toeplitz operators whose generating functions belong to the space of semi-almost periodic matrix functions.

This talk is based on common work with Y. Karlovich and B. Silbermann.