Complex symmetry & weighted composition operators on fock spaces - Pham Viet Hai (2017)
The main results of the thesis lie at the intersection of three areas: dynamical systems, complex symmetric operators, and weighted composition operators. We introduce two new concepts: weighted composition conjugations in operator theory, and complex symmetric C_0-semigroups (C_0-groups) in dynamical systems. With the techniques of weighted composition operators, we solve completely the following problems on the Fock spaces F^2(C^n): - the description of weighted composition operators which are conjugations; - the criteria for bounded weighted composition operators to be complex symmetric. For complex symmetric C_0-semigroups, we prove a new version of Stone’s theorem: - if each element of a C_0-semigroup is C-symmetric with respect to a fixed conjugation C, then the generator is C-selfadjoint as an unbounded operator; - and vice-versa, if the generator is C-selfadjoint, then this C_0-semigroup is complex symmetric with respect to the conjugation C. More interestingly, we show that the class of complex symmetric C_0-groups contains unitary groups as a very particular case. Furthermore, we investigate this concept on the Fock space F^2(C) by making use of semigroups of weighted composition operators, and show that this a really generalization of unitary groups.