## 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.

### MSC:

 47B33 Linear composition operators 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces
Full Text:

### References:

 [1] Bourdon, P.S.; Shapiro, J.H., Cyclic phenomena for composition operators, Memoirs amer. math. soc., 125, (1997) [2] P.S. Bourdon, Components of linear-fractional composition operators, J. Math. Anal. Appl., to appear · Zbl 1043.47021 [3] Brown, L.G.; Douglas, R.; Fillmore, P., Unitary equivalence modulo the compact operators and extensions of $$C\^{}\{∗\}$$-algebras, (), 58-128 [4] Cima, J.A.; Matheson, A., Essential norms of composition operators and Alexandrov measures, Pacific J. math., 179, 59-64, (1997) · Zbl 0871.47027 [5] Cowen, C.C., Linear fractional composition operators on H2, Integral equations operator theory, 11, 151-160, (1988) · Zbl 0638.47027 [6] Cowen, C.C., Composition operators on H2, J. operator theory, 9, 77-106, (1983) · Zbl 0504.47032 [7] Cowen, C.C.; MacCluer, B.D., Composition operators on spaces of analytic functions, (1995), CRC Press · Zbl 0873.47017 [8] Douglas, R.G., Banach algebra techniques in operator theory, (1972), Academic Press, 2nd ed., Springer, 1998 · Zbl 0247.47001 [9] Duren, P.L., Theory of Hp spaces, (1970), Academic Press · Zbl 0215.20203 [10] () [11] Littlewood, J.E., On inequalities in the theory of functions, Proc. London math. soc., 23, 481-519, (1925) · JFM 51.0247.03 [12] MacCluer, B.D., Components in the space of composition operators, Integral equations operator theory, 12, 725-738, (1989) · Zbl 0685.47027 [13] Nordgren, E.A., Composition operators, Canad. J. math., 20, 442-449, (1968) · Zbl 0161.34703 [14] Rudin, W., Real and complex analysis, (1987), McGraw-Hill · Zbl 0925.00005 [15] H.J. Schwartz, Composition operators on Hp, Thesis, University of Toledo, 1969 [16] Shapiro, J.H., The essential norm of a composition operator, Ann. math., 125, 375-404, (1987) · Zbl 0642.47027 [17] Shapiro, J.H.; Sundberg, C., Isolation amongst the composition operators, Pacific J. math., 145, 117-151, (1990) · Zbl 0732.30027 [18] Shapiro, J.H., Composition operators and classical function theory, (1993), Springer · Zbl 0791.30033 [19] Zorboska, N., Closed range essentially normal composition operators are normal, Acta sci. math. (Szeged), 65, 287-292, (1999) · Zbl 0938.47022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.