Automatic extensions of functional calculi

Ralph deLaubenfels

Studia Mathematica (1995)

  • Volume: 114, Issue: 3, page 237-259
  • ISSN: 0039-3223

Abstract

top
Given a Banach algebra ℱ of complex-valued functions and a closed, linear (possibly unbounded) densely defined operator A, on a Banach space, with an ℱ functional calculus we present two ways of extending this functional calculus to a much larger class of functions with little or no growth conditions. We apply this to spectral operators of scalar type, generators of bounded strongly continuous groups and operators whose resolvent set contains a half-line. For f in this larger class, one construction measures how far f(A) is from generating a strongly continuous semigroup, while the other construction measures how far f(A) is from being bounded. We apply our constructions to evolution equations.

How to cite

top

deLaubenfels, Ralph. "Automatic extensions of functional calculi." Studia Mathematica 114.3 (1995): 237-259. <http://eudml.org/doc/216190>.

@article{deLaubenfels1995,
abstract = {Given a Banach algebra ℱ of complex-valued functions and a closed, linear (possibly unbounded) densely defined operator A, on a Banach space, with an ℱ functional calculus we present two ways of extending this functional calculus to a much larger class of functions with little or no growth conditions. We apply this to spectral operators of scalar type, generators of bounded strongly continuous groups and operators whose resolvent set contains a half-line. For f in this larger class, one construction measures how far f(A) is from generating a strongly continuous semigroup, while the other construction measures how far f(A) is from being bounded. We apply our constructions to evolution equations.},
author = {deLaubenfels, Ralph},
journal = {Studia Mathematica},
keywords = {functional calculus; spectral operators of scalar type; generators of bounded strongly continuous groups; evolution equations},
language = {eng},
number = {3},
pages = {237-259},
title = {Automatic extensions of functional calculi},
url = {http://eudml.org/doc/216190},
volume = {114},
year = {1995},
}

TY - JOUR
AU - deLaubenfels, Ralph
TI - Automatic extensions of functional calculi
JO - Studia Mathematica
PY - 1995
VL - 114
IS - 3
SP - 237
EP - 259
AB - Given a Banach algebra ℱ of complex-valued functions and a closed, linear (possibly unbounded) densely defined operator A, on a Banach space, with an ℱ functional calculus we present two ways of extending this functional calculus to a much larger class of functions with little or no growth conditions. We apply this to spectral operators of scalar type, generators of bounded strongly continuous groups and operators whose resolvent set contains a half-line. For f in this larger class, one construction measures how far f(A) is from generating a strongly continuous semigroup, while the other construction measures how far f(A) is from being bounded. We apply our constructions to evolution equations.
LA - eng
KW - functional calculus; spectral operators of scalar type; generators of bounded strongly continuous groups; evolution equations
UR - http://eudml.org/doc/216190
ER -

References

top
  1. [1] M. Balabane, H. Emamirad and M. Jazar, Spectral distributions and generalization of Stone's theorem to the Banach space, Acta Appl. Math. 31 (1993), 275-295. Zbl0802.47013
  2. [2] G. Da Prato, Semigruppi regolarizzabili, Ricerche Mat. 15 (1966), 223-248. 
  3. [3] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980. Zbl0457.47030
  4. [4] R. deLaubenfels, Unbounded holomorphic functional calculus for operators with polynomially bounded resolvents, J. Funct. Anal. 114 (1993), 348-394. Zbl0785.47018
  5. [5] R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, Lecture Notes in Math. 1570, Springer, 1994. Zbl0811.47034
  6. [6] H. R. Dowson, Spectral Theory of Linear Operators, Academic Press, 1978. Zbl0384.47001
  7. [7] N. Dunford and J. T. Schwartz, Linear Operators, Part III, Interscience, New York, 1971. 
  8. [8] E. Marschall, Functional calculi for closed linear operators in Banach spaces, Manuscripta Math. 35 (1981), 277-310. Zbl0496.47033
  9. [9] L. E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, Pa., 1975. Zbl0302.35003
  10. [10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. Zbl0207.13501

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.