# American Institute of Mathematical Sciences

November  2012, 11(6): 2291-2303. doi: 10.3934/cpaa.2012.11.2291

## The annulus as a K-spectral set

 1 Institut de Recherche Mathématique de Rennes, UMR no. 6625, Université de Rennes 1, Campus de Beaulieu, 35042 RENNES Cedex

Received  March 2010 Revised  March 2010 Published  April 2012

We consider the annulus $\mathcal{A}_R$ of complex numbers with modulus and inverse of modulus bounded by $R>1$. We present some situations, in which this annulus is a K-spectral set for an operator $A$, and some related estimates.
Citation: Michel Crouzeix. The annulus as a K-spectral set. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2291-2303. doi: 10.3934/cpaa.2012.11.2291
##### References:
 [1] K. Okubo and T. Ando, Constants related to operators of class $C_\rho$, Manuscripta Math., 16 (1975), 385-394. doi: 10.1007/BF01323467. [2] C. Badea, B. Beckermann and M. Crouzeix, Intersections of several disks of the Riemann sphere as $K$-spectral sets, Com. Pure Appl. Anal., 8 (2009), 37-54. [3] W. F. Donoghue, On a problem of Nieminen, Inst. Hautes Etudes Sci. Publ. Math., 16 (1963), 31-33. doi: 10.1007/BF02684290. [4] R. G. Douglas and V. I. Paulsen, Completely bounded maps and hypo-Dirichlet algebras, Acta Sci. Math. (Szeged), 50 (1986), 143-157. [5] J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr., 4 (1951), 258-281. [6] V. I. Paulsen, Toward a theory of $K$-spectral sets, in "Surveys of Some Recent Results in Operator Theory,'' Vol. I, 221-240, Pitman Res. Notes Math. Ser., 171, Longman Sci. Tech., Harlow, 1988. [7] V. I. Paulsen, "Completely Bounded Maps and Operator Algebras,'' Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, 2002. [8] V. I. Paulsen and D. Singh, Extensions of Bohr's inequality, Bull. London Math. Soc., 38 (2006), 991-999. doi: 10.1112/S0024609306019084. [9] A. L. Shields, Weighted shift operators and analytic function theory, in "Topics in Operator Theory," pp. 49-128. Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974. [10] J. G. Stampfli, Minimal range theorems for operators with thin spectra, Pacific Journal of Math., 23 (1967), 601-612.

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##### References:
 [1] K. Okubo and T. Ando, Constants related to operators of class $C_\rho$, Manuscripta Math., 16 (1975), 385-394. doi: 10.1007/BF01323467. [2] C. Badea, B. Beckermann and M. Crouzeix, Intersections of several disks of the Riemann sphere as $K$-spectral sets, Com. Pure Appl. Anal., 8 (2009), 37-54. [3] W. F. Donoghue, On a problem of Nieminen, Inst. Hautes Etudes Sci. Publ. Math., 16 (1963), 31-33. doi: 10.1007/BF02684290. [4] R. G. Douglas and V. I. Paulsen, Completely bounded maps and hypo-Dirichlet algebras, Acta Sci. Math. (Szeged), 50 (1986), 143-157. [5] J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr., 4 (1951), 258-281. [6] V. I. Paulsen, Toward a theory of $K$-spectral sets, in "Surveys of Some Recent Results in Operator Theory,'' Vol. I, 221-240, Pitman Res. Notes Math. Ser., 171, Longman Sci. Tech., Harlow, 1988. [7] V. I. Paulsen, "Completely Bounded Maps and Operator Algebras,'' Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, 2002. [8] V. I. Paulsen and D. Singh, Extensions of Bohr's inequality, Bull. London Math. Soc., 38 (2006), 991-999. doi: 10.1112/S0024609306019084. [9] A. L. Shields, Weighted shift operators and analytic function theory, in "Topics in Operator Theory," pp. 49-128. Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974. [10] J. G. Stampfli, Minimal range theorems for operators with thin spectra, Pacific Journal of Math., 23 (1967), 601-612.
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