Complex symmetric operators and inverse spectral problem for Hankel operators

1 Jul 2021, 10:05


Sergei Treil (Brown University)


Hankel operators are bounded operators on $\ell^{2}$ whose matrix is constant on diagonals orthogonal to the main one (i.e. its entries depend only on the sum of indices). Such operators connect many classical problems in complex analysis with problems in operator theory.

I’ll be discussing the inverse spectral problem for such operators, i.e. the problem of finding a Hankel operator with prescribed spectral data. For non-selfadjoint operators the theory of the so-called complex symmetric operators gives a convenient way to present such spectral data.

It was discovered by P. Gerard and S. Grellier that the spectral data of a compact Hankel operator $\Gamma$ and the reduced Hankel operator $\Gamma S$ (where $S$ is the forward shift in $\ell^2$) completely determine the Hankel operator $\Gamma$. This turns out to be the case for general Hankel operators as well, i.e. the map from Hankel operators to the spectral data of $\Gamma$ and of $\Gamma S$ is injective. But what about surjectivity?

In the talk I'll discuss some positive results, as well as some counterexamples. Connections with Clark measures play an important role in the investigation, and will be discussed.

The talk is based on a joint work with P. Gerard and A. Pushnitskii.

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Sergei Treil (Brown University)

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