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Friday, November 13, 2020

**Abstract:** For certain spaces of analytic functions in several complex variables, we will show that bounded linear operators that preserve shift-cyclic functions (i.e. Tf is shift-cyclic whenever f is shift-cyclic) are necessarily weighted composition operators. Examples of spaces for which this result holds true consist of the Hardy space, Dirichlet type spaces, Drury-Arveson space etc. In the case of Hardy spaces we will show that this condition is also sufficient. That is, all bounded weighted composition operators preserve cyclic functions in the Hardy spaces.