Complex Symmetric Composition Operators Research Articles

Complex Symmetric Composition Operators on the Hardy Space of the Unit Ball

Complex Symmetric Composition Operators on the Hardy Space of the Unit Ball

Interplay between complex symmetry and Koenigs eigenfunctions

We investigate the relationship between the complex symmetry of composition operators Cϕf=f∘ϕ induced on the classical Hardy space H2(D) by an analytic self-map ϕ of the open unit disk D and its Koenigs eigenfunction. A generalization of orthogonality known as conjugate-orthogonality will play a key role in this work. We show that if ϕ is a Schröder map (fixes a point a∈D with 0<|ϕ′(a)|<1) and σ is its Koenigs eigenfunction, then Cϕ is complex symmetric if and only if (σn)n∈N is complete and conjugate-orthogonal in H2(D). We study the conjugate-orthogonality of Koenigs sequences with some concrete examples. We use these results to show that commutants of complex symmetric composition operators with Schröder symbols consist entirely of complex symmetric operators.

Complex symmetric composition operators on the Newton space

A conjecture posed by MacDonald and Rosenthal states the composition operator Cφ acting on the Newton space N2(P) induced by φ(z)=mz+m−12 is bounded, where m∈(0,1). Our first result is proving that the aforementioned conjecture holds. We investigate complex symmetry of composition operators acting on the Newton space, which is different from that on the Hardy space. In addition, normality and co-isometry of composition operators are studied. Finally, we consider complex symmetric weighted composition operators.

Complex symmetric composition operators on weighted Hardy spaces

Let φ \varphi be an analytic self-map of the open unit disk D \mathbb {D} . We study the complex symmetry of composition operators C φ C_\varphi on weighted Hardy spaces induced by a bounded sequence. For any analytic self-map of D \mathbb {D} that is not an elliptic automorphism, we establish that if C φ C_{\varphi } is complex symmetric, then either φ ( 0 ) = 0 \varphi (0)=0 or φ \varphi is linear. In the case of weighted Bergman spaces A α 2 A^{2}_{\alpha } , we find the non-automorphic linear fractional symbols φ \varphi such that C φ C_{\varphi } is complex symmetric.

Complex symmetric composition operators induced by linear fractional maps

Complex symmetric composition operators induced by linear fractional maps

Complex symmetric composition operators on a Hilbert space of Dirichlet series

On the Hilbert space of Dirichlet series with square summable coefficients, there are no non-normal complex symmetric composition operators induced by non-constant symbols. This is in sharp contrast with the phenomenon on the classical Hardy space over the unit disk.

Complex symmetric composition operators on H2

In this paper, we find complex symmetric composition operators on the classical Hardy space H2 whose symbols are linear-fractional but not automorphic. In doing so, we answer a recent question of Noor, and partially answer the original problem posed by Garcia and Hammond.

Complex symmetry of weighted composition operators in several variables

In this paper, we investigate analytic symbols [Formula: see text] and [Formula: see text] when the weighted composition operator [Formula: see text] is complex symmetric on general function space [Formula: see text]. As applications, we characterize completely the compactness, normality and isometry of complex symmetric weighted composition operators. Especially, we show that the equivalence of compactness and Hilbert–Schmidtness, and the existence of non-normal complex symmetric operators for such operators, which answers one open problem raised by Noor in [On an example of a complex symmetric composition operators on [Formula: see text], J. Funct. Anal. 269 (2015) 1899–1901] for higher dimensional case.

Complex symmetry of invertible composition operators

A bounded operator T on a separable Hilbert space H is said to be complex symmetric if there exists an orthonormal basis for H with respect to which T has a self-transpose matrix representation. In this paper, we study the complex symmetry of composition operators Cϕf=f∘ϕ induced on the Hardy space H2 by holomorphic self-maps ϕ of the open unit disk D. For any holomorphic self-map ϕ of D, we establish that if Cϕ is complex symmetric, then ϕ must fix a point in D. Thus among the automorphisms of D, only the elliptic ones may induce complex symmetric composition operators. For an elliptic automorphism ϕ, we prove that if ϕ is not a rotation or of order 3, then Cϕ is complex symmetric if and only ifϕ(z)=α−z1−α¯z for some α∈D∖{0}.