Complex symmetric weighted composition operators on the Fock space in several variables

Abstract

We characterize all selfadjoint as well as all unitary anti-linear weighted composition operators acting on the Fock space F 2 (C n ). Then we obtain a complete description of anti-linear weighted composition operators that are conjugations. These results allow us to determine which bounded linear weighted composition operators can be complex symmetric on F 2 (C n ).

Complex symmetric weighted composition operators on Hγ(D)

“…The following result, mentioned in the context of the Fock space in [7], holds for any separable complex Hilbert space H. Proposition 1.1. A is a conjugation on H if and only if A is self-adjoint and unitary on H.…”

“…where f belongs to the Hardy space. Also, a criterion for the complex symmetric structure of W ψ,ϕ was discovered on Fock space F 2 (C) (see [6]) and F 2 (C n ) (see [7]) with respect to a more general conjugation of the form…”

Weighted composition operators between Fock spaces in several variables

“…[20]). From this reason they have been studied intensively in different directions (see, for instance, [1, 2, 11, 14] for Toeplitz operators, [16, 19] for Hankel operators, [5, 12] for Volterra‐type integration operators, [3, 4, 7–9, 13, 18] for (weighted) composition operators).…”

Complex symmetric weighted composition operators on H_gamma(D)

“…Also, Hai and Khoi discovered the complex symmetric structure of weighted composition operators on both Fock spaces F 2 (C) (see [8]) and F 2 (C n ) (see [9] and [10]) with respect to a more general conjugation of the form…”

“…In the abovementioned papers [4,8,9,10,11], the crux to discovering the various criterions for the weighted composition operator to be complex symmetric with respect to their respective conjugations and function spaces lies within the reproducing kernels of the respective function spaces in question. We postpone a more detailed explanation on reproducing kernels to the next chapter.…”