Rocky Mountain Journal of Mathematics

The $C^*$-algebra generated by irreducibleToeplitz and composition operators

Massoud Salehi Sarvestani and Massoud Amini

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We describe the $C^*$-algebra generated by an irreducible Toeplitz operator~$T_{\psi }$, with continuous symbol~$\psi $ on the unit circle $\mathbb {T}$, and finitely many composition operators on the Hardy space $H^2$ induced by certain linear fractional self-maps of the unit disc, modulo the ideal of compact operators $K(H^2)$. For specific automorphism-induced composition operators and certain types of irreducible Toeplitz operators, we show that the above $C^*$-al\-ge\-bra is not isomorphic to that generated by the shift and composition operators.

Article information

Rocky Mountain J. Math., Volume 47, Number 4 (2017), 1301-1316.

First available in Project Euclid: 6 August 2017

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Zentralblatt MATH identifier

Primary: 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) [See also 46E22] 47B33: Composition operators

$C^*$-algebra shift operator irreducible Toeplitz operator composition operator linear-fractional map automorphism of the unit disk


Sarvestani, Massoud Salehi; Amini, Massoud. The $C^*$-algebra generated by irreducibleToeplitz and composition operators. Rocky Mountain J. Math. 47 (2017), no. 4, 1301--1316. doi:10.1216/RMJ-2017-47-4-1301.

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  • P. Bourdon, D. Levi, S. Narayan and J. Shapiro, Which linear fractional composition operators are essentially normal?, J. Math. Anal. Appl. 280 (2003), 30–53.
  • A. Brown and P.R. Halmos, Algebraic properties of Toeplitz operators, J. reine angew. Math. 213 (1964), 89–102.
  • C. Carathéodory, Über die gegenseitige Beziehung der Ränder bei der konformen Abbildung des Inneren einer Jordanschen Kurve auf einen Kreis, Math. Ann. 73 (1913), 305–320.
  • L. Coburn, The $C^*$-algebra generated by an isometry I, Bull. Amer. Math. Soc. 73 (1967), 722–726.
  • ––––, The $C^*$-algebra generated by an isometry II, Trans. Amer. Math. Soc. 137 (1969), 211–217.
  • J. Gray, On the history of thr Riemann mapping theorem, Rend. Circ. Matem. 34 (1994), 47–94.
  • M.T. Jury, $C^*$-algebras generated by groups of composition operators, Indiana Univ. Math. J. 56 (2007), 3171–3192.
  • ––––, The Fredholm index for elements of Toeplitz-composition $C^*$-algebras, Int. Equat. Oper. Th. 58 (2007), 341–362.
  • T. Kriete, B. MacCluer and J. Moorhouse, Toeplitz-composition $C^*$-algebras, J. Operator Theory 58 (2007), 135–156.
  • T. Kriete, B. MacCluer and J. Moorhouse, Spectral theory for algebraic combinations of Toeplitz and composition operators, J. Funct. Anal. 257 (2009), 2378–2409.
  • ––––, Composition operators within singly generated composition $C^*$-algebras, Israel J. Math. 179 (2010), 449–477.
  • A. Montes-Rodríguez, M. Ponce-Escudero and S. Shkarin, Invariant subspaces of parabolic self-maps in the Hardy space, Math. Res. Lett. 17 (2010), 99–107.
  • G. Murphy, $C^{*}$-Algebras and operator theory, Academic Press, New York, 1990.
  • E.A. Nordgren, Reducing subspaces of analytic Toeplitz operators, Duke Math. J. 34 (1967), 175–182.
  • K.S. Quertermous, Fixed point composition and Toeplitz-composition $C^*$-\nobreak algebras, J. Funct. Anal. 265 (2013), 743–764.
  • J.H. Shapiro, Composition operators and classical function theory, Springer Verlag, Berlin, 1993.
  • D.P. Williams, Crossed products of C*-algebras, American Mathematical Society, Providence, RI, 2007.