COLLABORATION IN BASIC SCIENCE AND ENGINEERING (COBASE)

Ilya Spitkovsky hosted Sergei Grudsky of Rostov State University, Russia for two weeks in February 2000. The purpose of this visit was to discuss possible collaboration on research involving Toeplitz and Wiener-Hopf operators. Spitkovsky reports that not only were possibilities of future collaboration discussed, but also some results were obtained during this visit. The proposed research was approached as three separate projects. The first involving partial indices of left and right Wiener-Hopf factorizations has been completed. As a result of their work, Spitkovsky and Grudsky developed a full description of all possible tuples of both left and right partial indices in the case of arbitrary measurable matrix functions. They also showed that all possibilities in fact occur already in the class of semi almost periodic matrix functions, and they established exact bounds for the total indices of such matrix functions. Two papers on this research have been submitted for publication. The second topic on spectra of Toeplitz operators with semi almost periodic symbols and the third regarding application of Toeplitz operators to some problems in air acoustics are still in progress. A proposal for future funding for work on the second and third projects will be submitted to NSF in the future.