Complex Symmetric Weighted Composition Operators on the Hardy Space
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Czechoslovak Mathematical Journal
15 pp
Online first
COMPLEX SYMMETRIC WEIGHTED COMPOSITION OPERATORS ON THE HARDY SPACE Cao Jiang, Nanchang, Shi-An Han, Ze-Hua Zhou, Tianjin Received December 18, 2018. Published online March 23, 2020.
Abstract. This paper identifies a class of complex symmetric weighted composition operators on H 2 (D) that includes both the unitary and the Hermitian weighted composition operators, as well as a class of normal weighted composition operators identified by Bourdon and Narayan. A characterization of algebraic weighted composition operators with degree no more than two is provided to illustrate that the weight function of a complex symmetric weighted composition operator is not necessarily linear fractional. Keywords: complex symmetry; weighted composition operator; Hardy space MSC 2020 : 47B33, 47B38
1. Preliminaries 1.1. Complex symmetry. Let H be a complex Hilbert space and L(H) the collection of all continuous linear operators on H. A map C : H → H is called a conjugation over H if it is ⊲ anti-linear: C(ax + by) = a ¯C(x) + ¯bC(y), x, y ∈ H, a, b ∈ C; ⊲ isometric: kCxk = kxk, x ∈ H; ⊲ involutive: C 2 = I.
An operator T ∈ L(H) is called complex symmetric if CT = T ∗ C
for some conjugation C and in this case we say T is complex symmetric with conjugation C. This work has been supported in part by the National Natural Science Foundation of China (Grant Nos. 11771323, 11371276). DOI: 10.21136/CMJ.2020.0555-18
1
The general study of complex symmetric operators was started by Garcia, Putinar and Wogen in [7], [8], [9], [10]. The class of complex symmetric operators turns to be quite diverse, see [7], [8], [10]. In this paper, we focus on two classes: normal operators and algebraic operators with degree no more than two. The spectral theorem states that if a normal operator is unitarily equivalent to some multiplier Mφ : L2 (X, dv) → L2 (X, dv), then it is easy to check that Mφ is complex symmetric with the usual conjugation Cf (z) = f (z). An operator T ∈ L(H) is said to be algebraic if it is annihilated by some nonzero polynomial p and the minimal degree of p is called the degree of T . Garcia and Wogen in [10] proved that an algebraic operator with degree no more than two is complex symmetric. 1.2. Hardy space. The classical Hardy-Hilbert space is defined by H 2 (D) =
f ∈ H(D); kf k2 = sup
1 0 1, if we let z = 0, then we get ψ(ϕ(0))[ϕ(ϕ(0))]n = Bϕ(0)n , that is,
h ϕ(ϕ(0)) in ϕ(0)
=
B , ψ(ϕ(0))
n > 1.
So we must have ϕ(ϕ(0)) = ϕ(0) and B = ψ(ϕ(0)). Evaluating (2.4) at ϕ(0) yields ψ(ϕ(0))2 ϕn (ϕ(0)) = Bψ(ϕ(0))ϕn (ϕ(0)) + Cϕ(0)n = ψ(ϕ(0))2 ϕn (ϕ(0)) + Cϕ(0)n , so we get ϕ(0) = 0. It follows from (2.4) that (2.5)
ψ(z) · ψ(ϕ(z))
h ϕ(ϕ(z)) in z
= Bψ(z)
ϕ(z) n z
+ C,
n > 0.
Since ϕ(0) = 0, equation (2.5), along with the Schwartz lemma, imply that ϕ(z) = λz for some |λ| = 1. Now, equation (2.4) is equivalent to ψ(z)ψ(λz)λ2n = Bψ(z)λn + C,
(2.6)
n > 0.
If λ = 1, then ψ 2 (z) = Bψ(z) + C, where ψ has to be constant and then Wψ,ϕ is a multiple of the identity map
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