Compactness and decompositions in ergodic and operator theory

• Applicant: Professor Dr. Balint Farkas
• Subject Area: Mathematics
• Term: from 2020 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 451698284

The interplay between operator and ergodic theory has been of major significance since the inception of both mathematical disciplines. Two concepts have played a fundamental role for both fields and their connection: Decompositions and compactness.The classical mean ergodic theorem, inspired by the ergodic hypothesis of thermodynamics, describing the long-term behavior of linear contractions on Hilbert spaces is actually equivalent to the von Neumann decomposition of the Hilbert space. And the validity of the mean ergodic theorem for linear contractions on Banach spaces with basis is actually equivalent to the weak compactness of the unit ball, i.e., reflexivity. Other decomposition and compactness properties results can be used to prove several other “ergodic theorems” motivated by physics or, surprisingly, number theory. As in the mentioned cases, decomposition results are often related to compactness properties of operator semigroups. A key tool to deduce decompositions from compactness are so-called enveloping semigroups which also play an important role in the classification of “structured” dynamical systems: Given a map or an operator satisfying a suitable compactness condition, one considers the generated cyclic semigroup and its closure. For the resulting compact semigroups one can apply an elegant structure theory yielding impressive and deep results for dynamical systems and operator theory. One of the most prominent operator theoretic structure theorems is the one of Jacobs-de Leeuw-Glicksberg, presenting the decomposition of a weakly compact operator semigroup into a structured (compact group) and a random (chaotic, almost weakly stable) part.In our project we plan to focus on three aspects: First, we investigate criteria for compactness. What conditions on the Banach space or the operator (semigroup) ensure compactness properties? How are they related to other concepts such as mean ergodicity or mean almost periodicity of vector-valued functions on groups? Second, we strive to generalize decompositions and compactness to a “relative” setting: Using these concepts we develop a systematic operator theoretic approach to (structured) extensions of dynamical systems and introduce enveloping “semigroupoids” as a generalized version of enveloping semigroups. Having a proper representation theory for such enveloping semigroupoids will hopefully allow for a relativized version of a Jacobs-de Leeuw-Glicksberg type decomposition for Banach bundles. Third and finally, we apply known as well as our new results to solve concrete problems of operator theory and dynamical systems. In particular, we aim to study non-conventional (multiple and subsequential) ergodic theorems from the operator theoretic perspective, to find new ergodic theorems and investigate the periodic decomposition problem for functions on semigroups.

• DFG Programme: Research Grants