Functional Analysis and Applications Seminar 

15/01/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
Frank-Olme Speck, Instituto Superior Técnico, Universidade Técnica de Lisboa

On some connections between pure Hankel operators, extensions of Helmholtz solutions into conical Riemann surfaces and the factorization of a special matrix

Siegfried Proessdorf studied with me, in 1987-89, the factorization of Daniele-Khrapkov matrices in spaces of Hoelder continuous functions. The work was motivated by a special matrix function that was found in diffraction theory by E. Meister in 1977 and "ad hoc factored" by A.R. Rawlins in 1981. It was recognized soon as an important example of a matrix function that admits an explicit generalized factorization (although not rationally reducible to a triangular matrix), and gave a great impact on factorization theory, developed in our research center at Lisbon. Recently diffraction by non-rectangular (but rational) wedges has shown an interesting connection with the existence of certain extension operators, either from half-lines into cones bordered by this half-line on one side, or into cones which contain this half-line in its interior up to the common vertex, such that the extended function is a weak solution of the Helmholtz equation. The existence of the latter extension operator is somehow equivalent to the inversion of a certain pure Hankel operator. In this lecture we focus on the relations between the above-mentioned operators and formulate some resulting open problems.