Components of linear-fractional composition operators. (English) Zbl 1043.47021
Let \(\phi\) denote a holomorphic function on the open unit disc \(U\) in the complex plane with \(\phi(U) \subset U\). The composition operator \(C_{\phi}(f):=f \circ \phi\) is bounded in the Hardy space \(H^2\). Denote by \(\text{comp} (H^2)\) the collection of all composition operators on \(H^2\) endowed with the metric induced by the operator norm. J. H. Shapiro and C. Sundberg [Pac. J. Math. 145, 117–121 (1990; Zbl 0732.30027)], improving work by Berkson, studied isolated points in \(\text{comp} (H^2)\) and showed, e.g., that only extreme points of \(H^{\infty}(U)\) can induce isolated composition operators. They also conjectured that if two composition operators \(C_{\phi}\) and \(C_{\psi}\) belong to the same component of \(\text{comp} (H^2)\), then \(C_{\phi} - C_{\psi}\) is compact.
Two analytic self-maps \(\phi\) and \(\psi\) are said to have the same first order boundary data if, whenever \(\phi\) or \(\psi\) has a finite angular derivative at \(\zeta\) in the boundary of \(U\), then \(\phi\) and \(\psi\) have the same radial limit and the same angular derivative at \(\zeta\). B. D. MacCluer [Integral Equations Oper. Theory 12, 725–738 (1989; Zbl 0685.47027)] showed that if \(C_{\phi}\) and \(C_{\psi}\) belong to the same component of \(\text{comp} (H^2)\), then \(\phi\) and \(\psi\) have the same first order boundary data. The author proves the converse, if both \(\phi\) and \(\psi\) are linear fractional, and even that \(C_{\phi}\) and \(C_{\psi}\) can be joined by a continuous path in \(\text{comp} (H^2)\).
In the final section, the author proves that, for analytic self-maps \(\phi\) and \(\psi\) which extend to be \(C^2\) on the closed unit disc, a necessary condition for \(C_{\phi}- C_{\psi}\) to be compact is that \(\phi\) and \(\psi\) have the same second order boundary data, thus obtaining an example disproving the conjecture of Shapiro and Sundberg. He also shows that the component of a non-automorphic, linear fractional self-map \(\phi\) on \(U\) always contains composition operators whose symbols are not linear fractional. In a remark, the author points out that J. Moorhouse and C. Toews [Contemp. Math. 321, 207-213 (2003; Zbl 1052.47018)] were the first to disprove the conjecture of Shapiro and Sundberg.
Two analytic self-maps \(\phi\) and \(\psi\) are said to have the same first order boundary data if, whenever \(\phi\) or \(\psi\) has a finite angular derivative at \(\zeta\) in the boundary of \(U\), then \(\phi\) and \(\psi\) have the same radial limit and the same angular derivative at \(\zeta\). B. D. MacCluer [Integral Equations Oper. Theory 12, 725–738 (1989; Zbl 0685.47027)] showed that if \(C_{\phi}\) and \(C_{\psi}\) belong to the same component of \(\text{comp} (H^2)\), then \(\phi\) and \(\psi\) have the same first order boundary data. The author proves the converse, if both \(\phi\) and \(\psi\) are linear fractional, and even that \(C_{\phi}\) and \(C_{\psi}\) can be joined by a continuous path in \(\text{comp} (H^2)\).
In the final section, the author proves that, for analytic self-maps \(\phi\) and \(\psi\) which extend to be \(C^2\) on the closed unit disc, a necessary condition for \(C_{\phi}- C_{\psi}\) to be compact is that \(\phi\) and \(\psi\) have the same second order boundary data, thus obtaining an example disproving the conjecture of Shapiro and Sundberg. He also shows that the component of a non-automorphic, linear fractional self-map \(\phi\) on \(U\) always contains composition operators whose symbols are not linear fractional. In a remark, the author points out that J. Moorhouse and C. Toews [Contemp. Math. 321, 207-213 (2003; Zbl 1052.47018)] were the first to disprove the conjecture of Shapiro and Sundberg.
Reviewer: José Bonet (Valencia)
MSC:
| 47B33 | Linear composition operators |
References:
| [1] | Berkson, E., Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc., 81, 230-232 (1981) · Zbl 0464.30027 |
| [2] | Bourdon, P. S.; Shapiro, J. H., Cyclic phenomena for composition operators, Mem. Amer. Math. Soc., 596, 1-105 (1997) · Zbl 0996.47032 |
| [3] | P.S. Bourdon, D. Levi, S. Narayan, J.H. Shapiro, Which linear fractional composition operators are essentially normal?, J. Math. Anal. Appl., to appear; P.S. Bourdon, D. Levi, S. Narayan, J.H. Shapiro, Which linear fractional composition operators are essentially normal?, J. Math. Anal. Appl., to appear · Zbl 1024.47008 |
| [4] | Cowen, C. C.; MacCluer, B. D., Composition Operators on Spaces of Analytic Functions (1995), CRC Press: CRC Press Boca Raton · Zbl 0873.47017 |
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| [6] | Littlewood, J. E., On inequalities in the theory of functions, Proc. London Math. Soc., 23, 481-519 (1925) · JFM 51.0247.03 |
| [7] | MacCluer, B. D., Components in the space of composition operators, Integral Equations Operator Theory, 12, 725-738 (1989) · Zbl 0685.47027 |
| [8] | MacCluer, B. D.; Shapiro, J. H., Angular derivatives and compact composition operators on Hardy and Bergman spaces, Canad. J. Math., 38, 878-906 (1986) · Zbl 0608.30050 |
| [9] | J. Moorhouse, C. Toews, Differences of composition operators, in: Proceedings of the Conference “Trends in Banach Spaces,” Memphis, 2001, to appear; J. Moorhouse, C. Toews, Differences of composition operators, in: Proceedings of the Conference “Trends in Banach Spaces,” Memphis, 2001, to appear · Zbl 1052.47018 |
| [10] | Shapiro, J. H.; Sundberg, C., Isolation amongst the composition operators, Pacific J. Math., 145, 117-151 (1990) · Zbl 0732.30027 |
| [11] | Shapiro, J. H., Composition Operators and Classical Function Theory (1993), Springer-Verlag · Zbl 0791.30033 |
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