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題目一：  GROTHENDIECK-LIDSKII FORMULAE IN HYPERCOMPLEX ANALYSIS

摘要: The classic Grothendieck-Lidskii formula provides a connection between the trace and Fredholm determinant and the spectrum of a Fredholm operator. Unfortunately, linear algebra over non-commutative structures like Quaternions and Clifford algebras is quite different from classic linear algebra and, consequently, the classic formula does not hold in this case. In this talk we will discuss the difficulties and

show how a type of Grothendieck-Lidskii formula can be established in these cases.

題目二：Triangular decompositions of quaternionic non-self-adjoint operators

摘要: One of the principal problems in studying spectral theory for quaternionic or Clifford-algebra-valued operators lies in the fact that due to the noncommutativity many methods from classic spectral theory are not working anymore in this setting. For instance, even in the simplest case of finite rank operators there are different notions of a left and right spectrum. Hereby, the notion of a left spectrum has little practical use while the notion of a right spectrum is based on a nonlinear eigenvalue problem. In the present talk we will recall the notion of S-spectrum as a natural way to consider a spectrum in a noncommutative setting and use it to study quaternionic non-selfadjoint operators. To this end we will discuss quaternionic Volterra operators and triangular representation of quaternionic operators similar to the classic approaches by Gohberg, Krein, Livsic, Brodskii and de Branges. Hereby we introduce spectral integral representations with respect to quaternionic chains and discuss the concept of P- triangular operators in the quaternionic setting. This will allow us to study the localization of spectra of non-selfadjoint quaternionic operators and presented triangular decompositions of non-selfadjoint operators with respect to maximal quaternionic eigenchains.