Abstract
The paper provides new characterisations of generators of cosine functions and C 0-groups on UMD spaces and their applications to some classical problems in cosine function theory. In particular, we show that on UMD spaces, generators of cosine functions and C 0-groups can be characterised by means of a complex inversion formula. This allows us to provide a strikingly elementary proof of Fattorini’s result on square root reduction for cosine function generators on UMD spaces. Moreover, we give a cosine function analogue of McIntosh’s characterisation of the boundedness of the H ∞ functional calculus for sectorial operators in terms of square function estimates. Another result says that the class of cosine function generators on a Hilbert space is exactly the class of operators which possess a dilation to a multiplication operator on a vector-valued L 2 space. Finally, we prove a cosine function analogue of the Gomilko-Feng-Shi characterisation of C 0-semigroup generators and apply it to answer in the affirmative a question by Fattorini on the growth bounds of perturbed cosine functions on Hilbert spaces.
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References
Amann H.: Linear and Quasilinear Parabolic Problems. Berlin, Basel (1995)
Arendt W., BattyC. J. K., Hieber M., Neubrander F.: Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, vol. 96. Birkhäuser, Basel (2001)
Badea C., Crouzeix M., Delyon B.: Convex domain and K-spectral sets. Math. Z. 252, 345–365 (2006)
Batty C. J. K.: On a perturbation theorem of Kaiser and Weis. Semigroup Forum 70, 471–474 (2005)
Beckermann B., Crouzeix M.: Operators with numerical range in a conic domain. Arch. Math. 88, 547–559 (2007)
Böttcher, A. and Karlovich, Y., Carleson curves, Muckenhaupt weights, and Toeplitz operators, Birkhäuser, 1997.
van Casteren J.A.: Operators similar to unitary or selfadjoint ones. Pacific J. Math. 104, 241–255 (1983)
Chojnacki W.: On group decompositions of bounded cosine sequences. Studia Math. 181, 61–85 (2007)
Cioranescu I., Keyantuo V.: On operator cosine functions inUMDspaces. Semigroup Forum 63, 429–440 (2001)
Cioranescu I., Lizama C.: On the inversion of the Laplace transform for resolvent families in UMD spaces. Arch. Math. 81, 182–192 (2003)
Crouzeix M.: Operators with numerical range in a parabola. Arch. Math. 82, 517–527 (2004)
Crouzeix M.: Numerical range and functional calculus in Hilbert space. J. Funct. Anal. 244, 668–690 (2007)
Crouzeix M., Delyon B.: Some estimates for analytic functions of strip or sectorial operators. Arch. Math. 81, 559–566 (2003)
Driouich A., El-Mennaoui O.: On the inverse Laplace transform for C 0-semigroups in UMDspaces. Arch. Math. 72, 56–63 (1999)
Dunford, N. and Schwartz, J. T., Linear operators, part I, Interscience, New York, 1957.
Fattorini H. O.: Ordinary differential equations in linear topological spaces. I. J. Differential Equations 5, 72–105 (1969)
Fattorini H.O.: Ordinary differential equations in linear topological spaces. II. J. Differential Equations 6, 50–70 (1969)
Fattorini H.O.: Uniformly bounded cosine functions in Hilbert space. Indiana Univ. Math. J. 20, 411–425 (1970)
Fattorini H.O.: On the growth of solutions to second order differential equations in Banach spaces. Proc. Roy. Soc. Edinburgh Sect. A 101, 237–252 (1985)
Fattorini H.O.: Second Order Linear Differential Equation in Banach Spaces. North-Holland, Amsterdam (1985)
Feng D.X., Shi D.H.: Characteristic conditions of the generation of C 0-semigroups in a Hilbert space. J. Math. Anal. Appl. 247, 356–376 (2000)
Fröhlich A.M., Weis L.: H ∞ calculus and dilations. Bull. Soc. Math. France 134, 487–508 (2006)
Garnett, J. B., Bounded analytic functions. Revised first edition, Graduate Texts inMathematics, 236, Springer, New York, 2007.
Goldstein J.: The universal addability problem for generators of cosine functions and operator groups. Huston J. Math. 6, 365–373 (1980)
Gomilko A.M.: Conditions on the generators of a uniformly bounded C 0-semigroup. Funct. Anal. App. 33, 294–296 (1999)
Haase M.: Spectral properties of operator logarithms. Math. Z. 245, 761–779 (2003)
Haase M.: A characterisation of group generators on Hilbert spaces and the H ∞-calculus. Semigroup Forum 66, 288–304 (2003)
Haase, M., The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, Birkhäuser, Basel, 2006.
Haase M.: Functional calculus for groups and applications to evolution equations. J. Evol. Equ. 7, 529–554 (2007)
Haase M.: The complex inversion formula revisited. J. Aust.Math. Soc. 84, 73–83 (2008)
Haase M.: The group reduction for bounded cosine functions on UMD spaces. Math. Z. 262, 281–299 (2009)
Haase M.: A transference principle for general groups and functional calculus on UMD spaces. Math. Ann. 345, 245–265 (2009)
Hille E., Offord A.C., Tamarkin J.D.: Some observations on the theory of Fourier transforms. Bull. Amer. Math. Soc. 39, 768–774 (1935)
Hille, E., Phillips, R. S., Functional Analysis and Semigroups, Amer. Math. Soc., Providence, R.I., 1957.
Hille E., Tamarkin J.D.: On the theory of Fourier transforms. Bull. Amer. Math. Soc. 39, 768–774 (1933)
Kaiser C., Weis L.: A perturbation theorem for operator semigroups in Hilbert spaces. Semigroup Forum 67, 63–75 (2003)
Keyantuo V., Vieten P.: On analytic semigroups and cosine functions in Banach spaces. Studia Math. 129, 137–156 (1998)
Krein, G., Linear differential equations in Banach spaces (translated from the Russian by J.M. Danskin), Translations of Mathematical Monographs, Vol. 29, American Mathematical Society, Providence RI, 1971.
Kunstmann, P., Weis, L., Maximal L p -regularity for parabolic equations, Fourier multiplier theorems and H ∞ -functional calculus. In: Functional Analytic Methods for Evolution Equations. Lecture Notes in Math., vol. 1855, pp. 65–311. Springer, Berlin (2004).
Liu K.: A characterisation of strongly continuous groups of linear operators on a Hilbert space. Bull. London Math. Soc. 32, 54–62 (2000)
McIntosh, A., Operators which have a bounded H ∞ -calculus, Miniconference on Operator Theory and Partial Differential Equations 1986, Proceedings of the Centre for Mathematical Analysis ANU, Vol. 14, 210–231, 1986.
Miana P.J.: Vector-valued cosine transforms. Semigroup Forum 71, 119–133 (2005)
Putinar M., Sandberg S.: A skew normal dilation on the numerical range of an operator. Math. Ann. 331, 345–357 (2005)
Rubio de Francia, J., Martingale and integral transforms of Banach space valued functions, Probability and Banach spaces (Zaragoza, 1985), 195–222, Lecture Notes in Math., 1221, Springer, Berlin, 1986.
Titchmarsh, E. C., Introduction to the theory of Fourier integrals, Oxford 1948.
Vasil’ev V.V., Piskarev S.I.: Differential equations in Banach spaces .II. Theory of cosine operator functions. J. Math. Sci. 122, 3055–3174 (2004)
Zwart H.: On the invertibility and bounded extension of C 0-semigroups. Semigroup Forum 63, 153–160 (2001)
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The author was supported by Narodowe Centrum Nauki grant DEC-2011/03/B/ST1/00407.
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Król, S. Resolvent characterisation of generators of cosine functions and C 0-groups. J. Evol. Equ. 13, 281–309 (2013). https://doi.org/10.1007/s00028-013-0178-2
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DOI: https://doi.org/10.1007/s00028-013-0178-2