Complex symmetry & weighted composition operators on fock spaces
Pham Viet Hai
Date of Issue2017
School of Physical and Mathematical Sciences
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.