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A note on composition operators in a half-plane. (English) Zbl 1262.47035

The paper under review deals with composition operators on the Hardy space in the upper half-plane \[ {\mathbb C}^+=\{x+iy \in {\mathbb C} : y>0\}. \] An analytic function \(u : {\mathbb C}^+ \to {\mathbb C}\) belongs to the Hardy space \(H^2_{{\mathbb C}^+}\) when \[ \|u\|_2^2 = \sup_{y>0} \int_{-\infty}^\infty |u(x+iy)|^2\,dx. \] Given an analytic self-map \(\Phi\) of the upper half-plane, the authors consider the formal composition operator \(C_\Phi\) defined for every analytic function \(u\) on the upper half-plane by the expression \[ C_\Phi u = u \circ \Phi. \] Those symbols \(\Phi\) that induce a bounded composition operator on \(H^2_{{\mathbb C}^+}\) were characterized by V. Matache [Complex Anal. Oper. Theory 2, No. 1, 169–197 (2008; Zbl 1158.47019)]. The characterization goes as follows. First, write the map \(\Phi\) in the Nevanlinna form \[ \Phi(z)=\alpha + \beta z + \int_{-\infty}^\infty \frac{1+tz}{t-z}\,d\rho(t), \] where \(\alpha \in {\mathbb R},\) \(\beta \geq 0,\) and \(\rho\) is a finite, positive Borel measure on the real line \({\mathbb R}\). Then, \(\Phi\) is a bounded operator on \(H^2_{{\mathbb C}^+}\) if and only if \(\beta >0,\) in which case \(\|C_\Phi\|=\beta^{-1/2}\). The authors focus on the case where \(C_\Phi\) is a contraction, that is, \(\beta=1\). It is known that, in that case, \(C_\Phi\) is an isometry if and only if \(\rho\) is singular with respect to the Lebesgue measure. F. Bayart [Proc. Am. Math. Soc. 131, No. 6, 1789–1791 (2003; Zbl 1055.47020)] proved that, on the unit disk, all composition operators which are similar to isometries are induced by inner functions. S. Elliott [Oper. Matrices 6, No. 3, 503–510 (2012; Zbl 1298.47035), arXiv:1006.1987] proved an analogous result for the upper half-plane under the additional condition that the symbol is a rational function, and he conjectured that for an arbitrary symbol, the corresponding composition operator is not similar to an isometry unless it is already an isometry.
The authors show that there exist symbols \(\Phi\) for which \(\rho\) is absolutely continuous with respect to the Lebesgue measure (and, in particular, \(\Phi\) is not inner) and \(C_\Phi\) is similar to an isometry. They also provide criteria for \(C_\Phi\) to have closed range in terms of an associated family of probability measures on \({\mathbb R}\) which are the analogues of the so-called Alexandrov-Clark measures studied in the context of the unit disk.

MSC:

47B33 Linear composition operators
47A45 Canonical models for contractions and nonselfadjoint linear operators
30H15 Nevanlinna spaces and Smirnov spaces
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References:

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