Operator Classes

  • Karl E. Gustafson
  • Duggirala K. M. Rao
Part of the Universitext book series (UTX)

Abstract

The special properties of the numerical range of normal operators led to the creation of new operator classes. In particular, three classes of operators attracted a lot of attention. Each of these inherits one or more of the numerical range properties of the normal operator.

Keywords

Assure Hull Dition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes and References

Notes and References for Section 6.1

  1. G. Orland (1964). “On a Class of Operators,” Proc. Amer. Math. Soc. 15, 75–79MathSciNetMATHCrossRefGoogle Scholar
  2. V. Istratescu and I. Istratescu (1970). “On Bare and Semibare Points for Same Classes of Operators,” Portugaliae Mathematica 29, Fax 4, 205–211.MathSciNetMATHGoogle Scholar
  3. G. R. Luecke (1971). “A Class of Operators on Hilbert Space,” Pacific J. Math. 41, 153–156.MathSciNetGoogle Scholar
  4. T. Furuta (1977). “Relations between Generalized Growth Conditions and Several Classes of Convexoid Operators,” Canadian J. Math. 29, 1010–1030.MathSciNetMATHCrossRefGoogle Scholar

Notes and References for Section 6.2

  1. A. Wintner (1929). “Zur theorie beschränkten Bilinearformem,” Math. Z., Vol. 30, 228–282.MathSciNetMATHCrossRefGoogle Scholar
  2. T. Furuta (1973). “Some Characterizations of Convexoid Operators,” Rev. Roum. Math. Pures et Appl. 18, 893–900.MathSciNetMATHGoogle Scholar
  3. T. Furuta (1967). “On the Class of Paranormal Operators,” Proc. Japan Acad. 43, 594–598.MathSciNetMATHCrossRefGoogle Scholar
  4. T. Ando (unpublished, 1974).Google Scholar
  5. K. Gustafson and D. Rao (unpublished, 1985).Google Scholar
  6. D. K. Rao (1987). “Operadores Paranormales,” Revista Colombiana de Matematicas 21, 135–149.MathSciNetMATHGoogle Scholar
  7. J. G. Stampfli (1965). “Hyponormal Operators and Spectral Density,” Trans. Amer. Math. Soc. 117, 469–476.MathSciNetMATHCrossRefGoogle Scholar
  8. G. Lumer (1961). “Semi-inner-product Spaces,” Trans. Amer. Math. Soc. 100, 24–43.MathSciNetCrossRefGoogle Scholar
  9. G. Lumer and R. S. Phillips (1961). “Dissipative Operators in a Banach Space,” Pacific J. Math. 11, 679–698.MathSciNetMATHGoogle Scholar
  10. S. Prasanna (1981). “The Norm of a Derivation and the Björck—Thomee Istratescu Theorem,” Math Japonica 26, 585–588.MathSciNetMATHGoogle Scholar
  11. T. Furuta, S. Izumino and S. Prasanna (1982). “A Characterization of Centroid Operators,” Math. Japonica 27, 105–106.MathSciNetMATHGoogle Scholar

Notes and References for Section 6.3

  1. J. von Neumann (1951). “Eine Spektraltheorie für Allgemeine Operatoren eines Unitären Raumes,” Math. Nachr. 4, 258–281.MathSciNetMATHGoogle Scholar
  2. M. Schreiber (1963). “Numerical Range and Spectral Sets,” Michigan Math. J. 10, 283–288.MathSciNetMATHCrossRefGoogle Scholar
  3. S. Hildebrandt (1964). “The Closure of the Numerical Range as a Spectral Set,” Comm. Pure Appl. Math., 415–421.Google Scholar
  4. S. Hildebrandt (1966). “über den Numerischen Wertebereich eines Operators,” Math. Annalen 163, 230–247.MathSciNetMATHCrossRefGoogle Scholar
  5. K. Gustafson (1972). “Necessary and Sufficient Conditions for Weyl’s Theorem,” Michigan Math. J. 19, 71–81.MathSciNetMATHCrossRefGoogle Scholar
  6. A. Lebow (1963). “On Von Neumann’s Theory of Spectral Sets,” J. Math. Anal. Appl. 7, 64–90.MathSciNetMATHCrossRefGoogle Scholar
  7. T. Saito and T. Yoshino (1965). “On a Conjecture of Berberian,” Tohoku Math. J. 17, 147–149.MathSciNetMATHCrossRefGoogle Scholar
  8. B. A. Mirman (1968). “Numerical Range and Norm of a Linear Operator,” Trudy Seminara po Funkcional Analizu 10, 51–55.MathSciNetGoogle Scholar

Notes and References for Section 6.4

  1. S. K. Berberian (1970). “Some Conditions on an Operator Implying Normality,” Math. Ann. 184, 188–192.MathSciNetMATHCrossRefGoogle Scholar
  2. R. G. Douglas and P. Rosenthal (1968). “A Necessary and Sufficient Condition for Normality,” J. Math. Anal. Appl. 22, 10–11.MathSciNetMATHCrossRefGoogle Scholar
  3. J. G. Stampfli (1962). “Hyponormal Operators,” Pacific J. Math. 12, 1453–1458.MathSciNetMATHGoogle Scholar
  4. C. Meng (1963). “On the Numerical Range of an Operator,” Proc. Amer. Math. Soc. 14, 167–171.MathSciNetMATHCrossRefGoogle Scholar
  5. V. Istratescu and I. Istratescu (1967). “On Normaloid Operators,” Math. Zeitschr. 105, 153–156.MathSciNetCrossRefGoogle Scholar
  6. R. Grone, C. Johnson, E. Sa and H. Wolkowicz (1987). “Normal Matrices,” Lin. Alg. and Appl. 87, 213–225.MathSciNetMATHCrossRefGoogle Scholar

Notes and References for Section 6.5

  1. B. N. Moyls and M. D. Marcus (1955). “Field Convexity of a Square Matrix,” Proc. Amer. Math. Soc. 6, 981–983.MathSciNetMATHCrossRefGoogle Scholar
  2. M. Goldberg and G. Zwas (1974). “On Matrices Having Equal Spectral Radius and Spectral Norm,” Lin. Alg. Appl. 8, 427–434.MathSciNetMATHCrossRefGoogle Scholar
  3. M. Goldberg, E. Tadmore and G. Zwas (1975). “The Numerical Radius and Spectral Matrices,” Lin. Multilin. Alg. 2, 317–326.CrossRefGoogle Scholar
  4. M. Goldberg and G. Zwas (1976). “Inclusion Relations Between Certain Sets of Matrices,” Lin. Multilin. Alg. 4, 55–60.MathSciNetCrossRefGoogle Scholar

Notes and References for Section 6.6

  1. K. Gustafson (1996). “Trigonometric Interpretation of Iterative Methods,” Proc. Conf. on Algebraic Multilevel Iteration Methods with Applications, (O. Axelsson, ed.), June 13–15, Nijmegen, Netherlands, 23–29.Google Scholar
  2. K. Gustafson (1996). “Operator Spectral States,” Computers Math. Applic. to appear.Google Scholar
  3. K. Gustafson and D. Rao (1996). “Spectral States of Normal-like Operators,” to appear.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1997

Authors and Affiliations

  • Karl E. Gustafson
    • 1
  • Duggirala K. M. Rao
    • 2
  1. 1.Department of MathematicsUniversity of Colorado at BoulderBoulderUSA
  2. 2.Department of MathematicsColegio Bolivar at CaliCaliColombia

Personalised recommendations