Complex symmetry & weighted composition operators on fock spaces
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 dynamic...
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| Format: | Theses and Dissertations |
| Language: | English |
| Published: |
2017
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| Online Access: | http://hdl.handle.net/10356/72686 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | 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. |
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