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

The annulus as a K-spectral set

Michel Crouzeix 1,

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.
Keywords: numerical radius, spectral set, Numerical range.
Mathematics Subject Classification: Primary: 47A11; Secondary: 47A25.
Citation: Michel Crouzeix. The annulus as a K-spectral set. Communications on Pure & 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.  Google Scholar

[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.  Google Scholar

[3]

W. F. Donoghue, On a problem of Nieminen, Inst. Hautes Etudes Sci. Publ. Math., 16 (1963), 31-33. doi: 10.1007/BF02684290.  Google Scholar

[4]

R. G. Douglas and V. I. Paulsen, Completely bounded maps and hypo-Dirichlet algebras, Acta Sci. Math. (Szeged), 50 (1986), 143-157.  Google Scholar

[5]

J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr., 4 (1951), 258-281.  Google Scholar

[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.  Google Scholar

[7]

V. I. Paulsen, "Completely Bounded Maps and Operator Algebras,'' Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, 2002.  Google Scholar

[8]

V. I. Paulsen and D. Singh, Extensions of Bohr's inequality, Bull. London Math. Soc., 38 (2006), 991-999. doi: 10.1112/S0024609306019084.  Google Scholar

[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.  Google Scholar

[10]

J. G. Stampfli, Minimal range theorems for operators with thin spectra, Pacific Journal of Math., 23 (1967), 601-612.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

W. F. Donoghue, On a problem of Nieminen, Inst. Hautes Etudes Sci. Publ. Math., 16 (1963), 31-33. doi: 10.1007/BF02684290.  Google Scholar

[4]

R. G. Douglas and V. I. Paulsen, Completely bounded maps and hypo-Dirichlet algebras, Acta Sci. Math. (Szeged), 50 (1986), 143-157.  Google Scholar

[5]

J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr., 4 (1951), 258-281.  Google Scholar

[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.  Google Scholar

[7]

V. I. Paulsen, "Completely Bounded Maps and Operator Algebras,'' Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, 2002.  Google Scholar

[8]

V. I. Paulsen and D. Singh, Extensions of Bohr's inequality, Bull. London Math. Soc., 38 (2006), 991-999. doi: 10.1112/S0024609306019084.  Google Scholar

[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.  Google Scholar

[10]

J. G. Stampfli, Minimal range theorems for operators with thin spectra, Pacific Journal of Math., 23 (1967), 601-612.  Google Scholar

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