# Automatic extensions of functional calculi

Studia Mathematica (1995)

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

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topdeLaubenfels, 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

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- [4] R. deLaubenfels, Unbounded holomorphic functional calculus for operators with polynomially bounded resolvents, J. Funct. Anal. 114 (1993), 348-394. Zbl0785.47018
- [5] R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, Lecture Notes in Math. 1570, Springer, 1994. Zbl0811.47034
- [6] H. R. Dowson, Spectral Theory of Linear Operators, Academic Press, 1978. Zbl0384.47001
- [7] N. Dunford and J. T. Schwartz, Linear Operators, Part III, Interscience, New York, 1971.
- [8] E. Marschall, Functional calculi for closed linear operators in Banach spaces, Manuscripta Math. 35 (1981), 277-310. Zbl0496.47033
- [9] L. E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, Pa., 1975. Zbl0302.35003
- [10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. Zbl0207.13501

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