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**Which linear-fractional composition operators are essentially normal?**
*(English)*
Zbl 1024.47008

Let \(\varphi\) be a holomorphic map of the unit circle \(\mathbb{U}\) into itself and \(C_\varphi\) the composition operator on the Hardy space \(H^2=H^2(\mathbb{U})\) defined by \(C_\varphi f(z)=f[\varphi(z)]\). This interesting paper deals with the property of the operator \(C_\varphi\) to be nontrivially essentially normal. (An operator \(T\) on a Hilbert space is said to be essentially normal if the commutator \(TT^\ast - T^\ast T\) is compact, and nontrivially essentially normal if it is essentially normal, but neither normal nor compact).

It was known that a composition operator on \(H^2\) is normal if and only it is generated by a dilatation \(\varphi(z)=az\) with \(|a|\leq 1\) [cf. H. J. Schwartz, “Composition Operators on \(H^p\)” (Thesis, University of Toledo) (1969)]. Recently it was shown that among the conformal automorphisms of \(\mathbb{U}\) the rotations are the only ones that induce essentially normal composition operators on \(H^2\). In this paper, the authors characterize the nontrivially essentially normal composition operators in the following Theorem. A composition operator \(C_\varphi\) induced on \(H^2\) by a linear-fractional map \(\varphi\) is nontrivially essentially normal if and only if \(\varphi\) is a parabolic nonautomorphism.

It was known that a composition operator on \(H^2\) is normal if and only it is generated by a dilatation \(\varphi(z)=az\) with \(|a|\leq 1\) [cf. H. J. Schwartz, “Composition Operators on \(H^p\)” (Thesis, University of Toledo) (1969)]. Recently it was shown that among the conformal automorphisms of \(\mathbb{U}\) the rotations are the only ones that induce essentially normal composition operators on \(H^2\). In this paper, the authors characterize the nontrivially essentially normal composition operators in the following Theorem. A composition operator \(C_\varphi\) induced on \(H^2\) by a linear-fractional map \(\varphi\) is nontrivially essentially normal if and only if \(\varphi\) is a parabolic nonautomorphism.

Reviewer: Stefan G.Samko (Faro)

### MSC:

47B33 | Linear composition operators |

46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |

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\textit{P. S. Bourdon} et al., J. Math. Anal. Appl. 280, No. 1, 30--53 (2003; Zbl 1024.47008)

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