COMPLEX SYMMETRY OF TOEPLITZ OPERATORS AND COMPOSITION OPERATORS

COMPLEX SYMMETRY OF TOEPLITZ OPERATORS AND COMPOSITION OPERATORS

Denote by H_γ(𝔻) the Hilbert space of holomorphic functions over the open unit disk 𝔻 with the reproducing kernel $K^{\gamma}_{\omega}(z)={\frac{1}{(1-{\bar{\omega}}z)^{\gamma}}}$, γ > 0. Let $J_{\alpha}:J_{\alpha}f(z)={\overline{f({\overline{{\alpha}z})}}}$ for z ∈ 𝔻 and 𝛼 ∈ 𝜕𝔻, and let W_u,v be the weighted composition operator. In this paper, we prove that if b ∈ ℝ and $b{\neq}{\frac{k{\pi}}{2}}$, for any integer k, then (cos b+i sin bW_u,v)J_𝛼 is a conjugation if and only if $v(z)={\frac{c-z}{1-{\bar{c}}z}}$, u(z) = ±k^(γ)_c(z), c ∈ 𝔻 and ${\bar{c}}\,=\,c{\alpha}$, where k^(γ)_c(z) is the normalized reproducing kernel, or (cos b + i sin bW_u,v)J_𝛼 = λJ_𝛼 for some λ ∈ ℂ with |λ| = 1. We then derive a necessary and sufficient condition for a Toeplitz operator to be complex symmetric with respect to the conjugation (cos b + i sin bW_u,v)J_𝛼 on H_γ(𝔻) with γ ≥ 1. Similarly, we also conduct a similar study for composition operators on H_γ(𝔻) with γ > 0.