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Some Complex Symmetric Operators on Weighted Hardy Spaces
Periodical of Ocean University of China 2025, 55(9): 158-164
Published: 01 September 2025
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In this paper, we study the complex symmetry of weighted composite operators and Toeplitz operators on the weighted Hardy space H2(β) by constructing conjugate-linear involutions. When the weight coefficients are βn=bn, b≥1, we combine the operator J: J f ( z ) = f ( z ¯ ) ¯ with the bounded weighted composite operator Wu, v to give Wu, vJ, and characterize when it is a conjugate-linear isometric involution. Furthermore, we obtain a necessary and sufficient condition for when the bounded weighted composite operator Wψ, φ, with respect to Wu, vJ, is complex symmetric. On the classical Hardy space, i.e. βn≡1, we use the idea of permutation method to construct an involutional linear isometric operator Cσ, and characterize the complex symmetry of Toeplitz operators and weighted composite operators with respect to Cσ.

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