Complex symmetric Toeplitz operators on the generalized derivative Hardy space

• Published in: Journal of inequalities and applications 2022-06, Vol.2022 (1), p.1-12, Article 74
• Main Authors: Ko, Eungil, Lee, Ji Eun, Lee, Jongrak
• Format: Article
• Language: English
• Subjects: Analysis; Applications of Mathematics; Conjugation; Hilbert space; Mathematical analysis; Mathematics; Mathematics and Statistics; Operators (mathematics)
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-022-02810-3

The generalized derivative Hardy space S α , β 2 ( D ) consists of all functions whose derivatives are in the Hardy and Bergman spaces as follows: for positive integers α , β , S α , β 2 ( D ) = { f ∈ H ( D ) : ∥ f ∥ S α , β 2 2 = ∥ f ∥ H 2 2 + α + β α β ∥ f ′ ∥ A 2 2 + 1 α β ∥ f ′ ∥ H 2 2 < ∞ } , where H ( D ) denotes the space of all functions analytic on the open unit disk D . In this paper, we study characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space S α , β 2 ( D ) with respect to some conjugations C ξ , C μ , λ . Moreover, for any conjugation C , we consider the necessary and sufficient conditions for complex symmetric Toeplitz operators with the symbol φ of the form φ ( z ) = ∑ n = 1 ∞ φ ˆ ( − n ) ‾ z ‾ n + ∑ n = 0 ∞ φ ˆ ( n ) z n . Next, we also study complex symmetric Toeplitz operators with non-harmonic symbols on the generalized derivative Hardy space S α , β 2 ( D ) .