American Institute of Mathematical Sciences

Advanced
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

[1]

Katja Polotzek, Kathrin Padberg-Gehle, Tobias Jäger. Set-oriented numerical computation of rotation sets. Journal of Computational Dynamics, 2017, 4 (1&2) : 119-141. doi: 10.3934/jcd.2017004
[2]

Chaoqian Li, Yaqiang Wang, Jieyi Yi, Yaotang Li. Bounds for the spectral radius of nonnegative tensors. Journal of Industrial & Management Optimization, 2016, 12 (3) : 975-990. doi: 10.3934/jimo.2016.12.975
[3]

Massimiliano Guzzo, Giancarlo Benettin. A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 1-28. doi: 10.3934/dcdsb.2001.1.1
[4]

Vladimir Müller, Aljoša Peperko. On the Bonsall cone spectral radius and the approximate point spectrum. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5337-5354. doi: 10.3934/dcds.2017232
[5]

Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22
[6]

Rui Zou, Yongluo Cao, Gang Liao. Continuity of spectral radius over hyperbolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3977-3991. doi: 10.3934/dcds.2018173
[7]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169
[8]

Vladimir Müller, Aljoša Peperko. Lower spectral radius and spectral mapping theorem for suprema preserving mappings. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4117-4132. doi: 10.3934/dcds.2018179
[9]

Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390
[10]

Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347
[11]

Zeyu Xia, Xiaofeng Yang. A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3749-3763. doi: 10.3934/dcdsb.2020089
[12]

Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561
[13]

Chen Ling, Liqun Qi. Some results on $l^k$-eigenvalues of tensor and related spectral radius. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 381-388. doi: 10.3934/naco.2011.1.381
[14]

Victor Kozyakin. Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 143-158. doi: 10.3934/dcdsb.2010.14.143
[15]

Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447
[16]

Alan Elcrat, Ray Treinen. Numerical results for floating drops. Conference Publications, 2005, 2005 (Special) : 241-249. doi: 10.3934/proc.2005.2005.241
[17]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
[18]

Marx Chhay, Aziz Hamdouni. On the accuracy of invariant numerical schemes. Communications on Pure & Applied Analysis, 2011, 10 (2) : 761-783. doi: 10.3934/cpaa.2011.10.761
[19]

Michael Dellnitz, O. Junge, B Thiere. The numerical detection of connecting orbits. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 125-135. doi: 10.3934/dcdsb.2001.1.125
[20]

Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure & Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences