Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 781–800 | Cite as

Maximal Operators with Respect to the Numerical Range

  • Rosario CorsoEmail author


Let \(\mathfrak {n}\) be a nonempty, proper, convex subset of \(\mathbb {C}\). The \(\mathfrak {n}\)-maximal operators are defined as the operators having numerical ranges in \(\mathfrak {n}\) and are maximal with this property. Typical examples of these are the maximal symmetric (or accretive or dissipative) operators, the associated to some sesquilinear forms (for instance, to closed sectorial forms), and the generators of some strongly continuous semi-groups of bounded operators. In this paper the \(\mathfrak {n}\)-maximal operators are studied and some characterizations of these in terms of the resolvent set are given.


Numerical range Maximal operators Sector Strip Sesquilinear forms Strongly continuous semi-groups Cayley transform 

Mathematics Subject Classification

Primary 47A20 Secondary 47A12 47B44 47A07 



This work was supported by the project “Problemi spettrali e di rappresentazione in quasi *-algebre di operatori”, 2017, of the “National Group for Mathematical Analysis, Probability and their Applications” (GNAMPA – INdAM).


  1. 1.
    Ando, T., Nishio, K.: Positive selfadjoint extensions of positive symmetric operators. Tohoku Math. J. 22, 65–75 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arlinskiĭ, Y.M.: A class of contractions in a Hilbert space. Ukrain. Math. Zh. 39(6), 691–696 (1987). ((in Russian) English translation. in Ukr. Math. J. 39(6), 560–564 (1987))MathSciNetGoogle Scholar
  3. 3.
    Arlinskiĭ, Y.M.: Characteristic functions of operators of the class \(C(\alpha )\). Izv. Vyssh. Uchebn. Zaved. Mat. 2, 13–21 (1991). ((in Russian).English translation in Sov. Math., 35(2), 13-23 (1991))MathSciNetGoogle Scholar
  4. 4.
    Bade, W.G.: An operational calculus for operators with spectrum in a strip. Pac. J. Math. 3, 257–290 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Batty, C.J.K.: Unbounded operators: functional calculus, generation, perturbations. Extr. Math. 24(2), 99–133 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Batty, C.J.K.: Bounded \(H^\infty \)-calculus for strip-type operators. Integr. Equ. Oper. Theory 72, 159–178 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Corso, R., Trapani, C.: Representation theorems for solvable sesquilinear forms. Integr. Eq. Oper. Theory 89(1), 43–68 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Corso, R.: A Kato’s second type representation theorem for solvable sesquilinear forms. J. Math. Anal. Appl. 462(1), 982–998 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Corso, R.: A survey on solvable sesquilinear forms. In: Böttcher, A., Potts, D., Stollmann, P., Wenzel, D. (eds.) The Diversity and Beauty of Applied Operator Theory. Operator Theory: Advances and Applications, vol. 268, pp. 167–177. Birkhäuser, Cham (2018)CrossRefGoogle Scholar
  10. 10.
    Ćurgus, B.: On the regularity of the critical point infinity of definitizable operators. Integr. Equ. Oper. Theory 8, 462–488 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    DeLaubenfels, R.: Strongly continuous groups, similarity and numerical range on a Hilbert space. Taiwan. J. Math. 1(2), 127–33 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Di Bella, S., Trapani, C.: Some representation theorems for sesquilinear forms. J. Math. Anal. Appl. 451, 64–83 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics. Springer, Berlin (2000)zbMATHGoogle Scholar
  14. 14.
    Gustafson, K., Rao, D.: Numerical Range: The Field of Values of Linear Operators and Matrices. Springer, New York (1997)CrossRefGoogle Scholar
  15. 15.
    Haase, M.: The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, vol. 169. Birkhäuser, Basel (2006)CrossRefzbMATHGoogle Scholar
  16. 16.
    Halmos, P.R.: A Hilbert Space Problem Book, 2nd edn. Springer, New York (1982)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kolmanovich, V., Malamud, M.: An operator analog of the Schwarz–Levner lemma. Teor. Funkt. Funktsional. Anal. Prilozh. 48, 71–74 (1987). ((in Russian) English translation in J. Sov. Math., 49(2), 900–902 (1990))zbMATHGoogle Scholar
  19. 19.
    McIntosh, A.: Representation of bilinear forms in Hilbert space by linear operators. Trans. Am. Math. Soc. 131(2), 365–377 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    McIntosh, A.: Operators which have an \(H^\infty \) functional calculus. In: Jefferies, B., McIntosh, A., Ricker, W. (eds.) Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), pp. 210–231. Australian National University, Canberra (1986)Google Scholar
  21. 21.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  22. 22.
    Phillips, R.S.: Dissipative operators and hyperbolic systems of partial differential equations. Trans. Am. Math. Soc. 90, 193–254 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schmüdgen, K.: Unbounded Self-adjoint Operators on Hilbert Space. Springer, Dordrecht (2012)CrossRefzbMATHGoogle Scholar
  24. 24.
    Sebestyén, Z., Tarcsay, Z.: Characterizations of selfadjoint operators. Stud. Sci. Math. Hungar. 50, 423–435 (2013)MathSciNetzbMATHGoogle Scholar
  25. 25.
    von Neumann, J.: Allgemeine Eigenwertstheorie Hermitescher Funktionalopertoren. Math. Ann. 102, 49–131 (1929)CrossRefGoogle Scholar
  26. 26.
    Weidmann, J.: Linear Operators in Hilbert Spaces. Springer, Berlin (1980)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di PalermoPalermoItaly

Personalised recommendations