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Physical & Biological Sciences Division

Mathematics Department

Professor

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Regular Faculty

McHenry Library

McHenry Building Room #4118

Mathematics Department

Ph.D., Technische Universität Chemnitz, 1997

Torsten Ehrhardt's research interests range from functional analysis (in particular, operator theory and the theory of Banach algebras), over harmonic analysis and Wiener-Hopf factorization theory to complex analysis. His research is both motivated by applications in such areas as statistical physics and random matrix theory, as well as by questions arising in different areas of "pure'' mathematics.

One part of his research concerns the theory of Toeplitz operators. In 1997 he proved in his Ph.D. theses a conjecture raised by Fisher and Hartwig in 1968 in essentially all cases in which it can be expected to be true. This conjecture is a generalization of the strong Szegö-Widom limit theorem and concerns the asymptotic behavior of the determinants of Toeplitz matrices as the size of the matrices goes to infinity. Currently he is also analyzing the asymptotic behavior of the determinants of other structured matrices such as Toeplitz-plus-Hankel matrices. The interest in this topic arises in statistical physics (e.g., in connection with Ising models and dimer models) as well as random matrix theory (eigenvalue distributions for certain matrix ensembles).

Another part of his research is devoted to the spectral theory of structured matrices and convolution-type operators (such as Toeplitz, Hankel, Wiener-Hopf, and singular integral operators). The theory of Toeplitz and singular integral operators has intimate connections with Wiener-Hopf factorization theory. His research has shown that an invertibility theory can be established for a larger class of operators (namely, for Toeplitz-plus-Hankel and generalized singular integral operators) by embarking on a different kind of factorization theory. He has also worked on explicit factorization theory and is pursuing research on problems in abstract Fourier analysis on groups.

Torsten Ehrhardt is also interested in the analysis of stability of approximation methods for several classes of operators, which is of interest in numerical analysis. A very strong tool of investigation are Banach algebra methods including non-commutative Gelfand theory.

Another line of research Prof. Ehrhardt has been following for several years is the study of logarithmic residues of analytic Banach algebra valued functions. This is a simply stated, but non-trivial problem, at the conjunction of the areas of complex analysis, operator theory, and the theory of Banach algebras.

- T. Ehrhardt: Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel, Comm. Math. Phys. 262 (2006), 317–341.
- T. Ehrhardt: Invertibility theory for Toeplitz plus Hankel operators and singular integral operators with flip, J. Funct. Anal. 208 (2004), 64–106.
- E.L.Basor and T. Ehrhardt: Factorization theory for a class of Toeplitz + Hankel operators, J. Operator Theory 51 (2004), 411–433.
- T. Ehrhardt: Factorization theory for Toeplitz + Hankel operators and singular integral operator with flip, Habilitation theses, Technische Universität Chemnitz, (2004), 253 pages.
- H.Bart, T. Ehrhardt, and B.Silbermann: Logarithmic Residues in the Banach algebra generated by the compact operators and the identity, Math. Nachr 268 (2004), 3–30.

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