Skip to main content
SpringerLink
Log in

Taylor Functional Calculus

  • Reference work entry
  • First Online:
Operator Theory

  • 4417 Accesses

Abstract

The notion of spectrum of an operator is one of the central concepts of operator theory. It is closely connected with the existence of a functional calculus which provides important information about the structure of Banach space operators. The situation for commuting n-tuples of Banach space operators is much more complicated. There are many possible definitions of joint spectra. However, the joint spectrum introduced by J.L. Taylor has a distinguished property—there exists a functional calculus for functions analytic on a neighborhood of this spectrum. The present paper gives a survey of basic properties of the Taylor spectrum and Taylor functional calculus.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 999.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Albrecht, D.W.: Integral formulae for special cases of Taylor’s functional calculus. Stud. Math. 105, 51–68 (1993)

    MathSciNet  MATH  Google Scholar 

  2. Ambrozie, C.-G.: The Euler characteristic is stable under compact perturbations. Proc. Am. Math. Soc. 124, 2041–2050 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andersson, M.: Taylor’s functional calculus for commuting operators with Cauchy-Fantappie-Leray formulas. Int. Math. Res. Not. 6, 247–258 (1997)

    Article  Google Scholar 

  4. Chō, M., Takaguchi, M.: Boundary of Taylor’s joint spectrum for two commuting operators. Sci. Rep. Hirosaki Univ. 28, 1–4 (1981)

    MathSciNet  MATH  Google Scholar 

  5. Curto, R.E.: Connections between Harte and Taylor spectrum. Rev. Roumaine Math. Pures Appl. 31, 203–215 (1986)

    MathSciNet  MATH  Google Scholar 

  6. Curto, R.E.: Spectral theory of elementary operators. In: Elementary Operators and Applications (Blaubeuren 1991), pp. 3–52. World Scientific, River Edge (1992)

    Google Scholar 

  7. Eschmeier, J., Putinar, M.: Spectral Decompositions and Analytic Sheaves. London Mathematical Society Monographs New Series, vol. 10. Oxford University Press, Oxford (1996)

    Google Scholar 

  8. Frunzǎ, S.: The Taylor spectrum and spectral decompositions. J. Funct. Anal. 19, 390–421 (1975)

    Article  Google Scholar 

  9. Gunning, R.C., Rossi, H.: Analytic Functions of Several Complex Variables. Prentice Hall, Englewood Cliffs (1965)

    MATH  Google Scholar 

  10. Helemskii, A.Ya.: Homological methods in the holomorphic calculus of several operators in Banach spaces, after Taylor. Uspekhi Mat. Nauk 36, 127–172 (1981, in Russian). English translation: Russ. Math. Surv. 36 (1981)

    Google Scholar 

  11. Helemskii, A.Ya.: The Homology of Banach and Topological Algebras (translated from Russian). Mathematics and Its Applications (Soviet Series), vol. 41. Kluwer Academic, Dordrecht (1989)

    Google Scholar 

  12. Kordula, V., Müller, V.: Vasilescu-Martinelli formula for operators in Banach spaces. Stud. Math. 113, 127–139 (1995)

    MATH  Google Scholar 

  13. Levi, R.: Notes on the Taylor Joint Spectrum of Commuting Operators. Spectral Theory, Banach Center Publications, vol. 8, pp. 321–332. PWN – Polish Scientific Publishers, Warsaw (1982)

    Google Scholar 

  14. Müller, V.: The splitting spectrum differs from the Taylor spectrum. Stud. Math. 123, 291–294 (1997)

    MATH  Google Scholar 

  15. Müller, V.: Spectral Theory of Linear Operators, 2nd edn. Birkhäuser, Basel (2007)

    MATH  Google Scholar 

  16. Putinar, M.: The superposition principle for Taylor’s functional calculus. J. Oper. Theory 7, 149–155 (1982)

    MathSciNet  MATH  Google Scholar 

  17. Putinar, M.: Uniqueness of Taylor’s functional calculus. Proc. Am. Math. Soc. 89, 647–650 (1983)

    MathSciNet  MATH  Google Scholar 

  18. Słodkowski, Z.: An infinite family of joint spectra. Stud. Math. 61, 239–255 (1977)

    MATH  Google Scholar 

  19. Słodkowski, Z., Żelazko, W.: On joint spectra of commuting families of operators. Stud. Math. 50, 127–148 (1974)

    MATH  Google Scholar 

  20. Taylor, J.L.: A joint spectrum for several commuting operators. J. Funct. Anal. 6, 172–191 (1970)

    Article  MATH  Google Scholar 

  21. Taylor, J.L.: Analytic-functional calculus for several commuting operators. Acta Math. 125, 1–38 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vasilescu, F.-H.: Stability of index of a complex of Banach spaces. J. Oper. Theory 2, 247–275 (1979)

    MathSciNet  MATH  Google Scholar 

  23. Vasilescu, F.-H.: Analytic Functional Calculus and Spectral Decompositions. Editura Academiei Republicii Socialiste România, Bucuresti (1979, in Roumanian). English translation: D. Reidel, Dodrecht (1982)

    Google Scholar 

  24. Wrobel, V.: The boundary of Taylor’s joint spectrum for two commuting Banach space operators. Stud. Math. 84, 105–111 (1986)

    MathSciNet  MATH  Google Scholar 

  25. Zame, W.R.:, Existence uniqueness and continuity of functional calculus homomorphisms. Proc. Lond. Math. Soc. 39, 73–92 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  26. Żelazko, W.: Axiomatic approach to joint spectra I. Stud. Math. 64, 249–261 (1979)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Academy of Sciences of the Czech Republic, Prague, Czech Republic

    Vladimír Müller

Authors
  1. Vladimír Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimír Müller .

Editor information

Editors and Affiliations

  1. Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Daniel Alpay

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Basel

About this entry

Cite this entry

Müller, V. (2015). Taylor Functional Calculus. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_61

Download citation

Publish with us

Policies and ethics

Search

Navigation