Complex Symmetry of Invertible Composition Operators on Weighted Bergman Spaces
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Complex Analysis and Operator Theory
Complex Symmetry of Invertible Composition Operators on Weighted Bergman Spaces Osmar R. Severiano1 Received: 1 May 2020 / Accepted: 11 July 2020 © Springer Nature Switzerland AG 2020
Abstract In this article, we study the complex symmetry of composition operators Cφ f = f ◦φ induced on the weighted Bergman spaces A2β (D), by analytic self-maps of the unit disk. One of our main results shows that if Cφ is complex symmetric then φ must fix a point in D. From this, we prove that if φ is neither constant nor an elliptic automorphism of D and Cφ is complex symmetric then Cφ and Cφ∗ are cyclic operators. Moreover, by assuming φ is an elliptic automorphism of D which not a rotation and β ∈ N, we show that Cφ is not complex symmetric whenever φ has order greater than 2(3 + β). Keywords Complex symmetry · Composition operator · Disk automorphisms · Weighted Bergman spaces Mathematics Subject Classification 47B33 · 30H20 · 47B99
Introduction A bounded operator T on a separable Hilbert space H is complex symmetric if there exists an orthonormal basis for H with respect to which T has a self-transpose matrix representation. An equivalent definition also exists. A conjugate-linear operator C : H → H is said to be a conjugation if C 2 = I and C f , Cg = g, f for all f , g ∈ H. We say that T is C-symmetric if C T = T ∗ C, and complex symmetric if there exists a conjugation C with respect to which T is C-symmetric. The concept of complex symmetric operators on separable Hilbert spaces is a natural generaliza-
Communicated by Daniel Aron Alpay. This article is part of the topical collection “Linear Operators and Linear Systems” edited by Sanne ter Horst, Dmitry S. Kaliuzhnyi-Verbovetskyi and Izchak Lewkowicz.
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Osmar R. Severiano [email protected] IMECC, Campinas, Brazil 0123456789().: V,-vol
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tion of complex symmetric matrices, and their general study was initiated by Garcia, Putinar, and Wogen [9–12]. The class of complex symmetric operators contains a large number of concrete examples including all normal operators, binormal operators, Hankel operators, finite Toeplitz matrices, compressed shift operators, and the Volterra integral operator. If X is a Banach space of analytic functions in the open unit disk D and if φ is an analytic self-map of D, the composition operator with symbol φ is defined by Cφ f = f ◦ φ for f ∈ X . The emphasis here is on the comparison of the properties of Cφ with those of the symbol φ. For example, it is well known that Cφ is bounded on each weighted Bergman space A2β (D) [13]. The study of complex symmetry of composition operators on the Hardy space H 2 (D) was initiated by Garcia and Hammond in [8]. There they proposed the problem of characterizing all compositions operators that are complex symmetric on H 2 (D). They also showed that for each α ∈ D, the involutive automorphism of D φα (z) =
α−z 1 − αz
(0.1)
induces a non-normal complex symmetric composition operator. Another important work on complex
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