A short counterexample to the inverse generator problem on non-Hilbertian reflexive \(L^p\)-spaces. (English) Zbl 1359.47038

Let \(e^{tA} \) be a strongly continuous \(C_{0} \)-semigroup of Banach space operators. Suppose that its generator \(A\) is one-to-one and has dense range. Its inverse \(A^{-1} \) is also densely defined, closed, and with dense range. It is interesting to know whether \(A^{-1} \) also generates a \(C_{0} \) semigroup. The question is important, because in the case when the semigroup \(e^{tA} \) is analytical and bounded on some sector, the answer is positive, as \(A^{-1} \) is sectorial in the same sector.
A counterexample is known for the problem on \(L^{p} \) spaces with \(p>1\), \(p\neq 2\). The author constructs here a different counterexample for the same case. The short proof is based on Fourier analysis. As the author notes, the inverse generator problem remains open for semigroups on Hilbert spaces.


47D06 One-parameter semigroups and linear evolution equations
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[1] Arhancetcet, C.; Le Merdy, C., Dilation of Ritt operators on \(L\)\^{}{\(p\)}-spaces, Is- rael J. Math., 201, 373-414, (2014) · Zbl 1311.47042
[2] R. R. Coifman and G. Weiss, Transference methods in analysis. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 31. American Mathematical Society, Providence, R.I., 1976, pp. ii+59. · Zbl 0371.43009
[3] deLaubenfels, R., Inverses of generators of nonanalytic semigroups, Studia Math., 191, 11-38, (2009) · Zbl 1167.47035
[4] deLaubenfels, R., Inverses of generators, Proc. Amer. Math. Soc ., 104, 443-448, (1988) · Zbl 0692.47032
[5] K-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics, vol. 194. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. Springer-Verlag, New York, 2000, pp. xxii+586. · Zbl 1167.47035
[6] S. Fackler, Regularity Properties of Sectorial Operators: Counterexamples and Open Problems, Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Ed. by Wolfgang Arendt, Ralph Chill, and Yuri Tomilov, Vol. 250, Operator Theory: Advances and Applications, Birkhäuser Verlag, 2015, pp. x+496. · Zbl 1338.47045
[7] Fendler, G., Dilations of one parameter semigroups of positive ontra tions on \(L\)\^{}{\(p\)} spa es, Canad. J. Math., 49, 736-748, (1997) · Zbl 0907.47039
[8] Gomilko, O., Optimal estimates for the semigroup generated by the classical Volterra operator on \(L\)\^{}{\(p\)}-spaces, Semigroup Forum, 83, 343-350, (2011) · Zbl 1250.47044
[9] L. Grafakos, Classical Fourier analysis. Second. Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008, pp. xvi+489. · Zbl 1250.47044
[10] Gomilko, A.; Tomilov, Y.; Zwart, H., On the inverse operator of the generator of a C0-semigroup, Mat. Sb., 198, 35-50, (2007) · Zbl 1136.47029
[11] Komatsu, H., Fractional powers of operators, Paciifc J. Math., 19, 285-346, (1966) · Zbl 0154.16104
[12] P. C. Kunstmann and L. Weis. Maximal \(L\)_{\(p\)}-regularity for paraboli equations, Fourier multiplier theorems and \(H\)\^{}{∞}-functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 65-311. · Zbl 1097.47041
[13] C. Le Merdy, \(H\)\^{}{∞}-functional calculus and appli cations to maximal regularity. Semi-groupes d’opérateurs et calcul fonctionnel (Besançon, 1998), Publ. Math. UFR Sci. Tech. Besançon, vol. 16. Univ. Franche-Comté, Besançon, 1999, pp. 41-77. · Zbl 0154.16104
[14] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and os illa- tory integrals. Princeton Mathematical Series, vol. 43, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. Princeton University Press, Prince- ton, NJ, 1993, pp. xiv+695. · Zbl 1250.47044
[15] H. Zwart, Is \(A\)\^{}{1} an infinitesimal generator? Perspectives in operator theory. Ba- nach Center Publ. vol. 75, Polish A ad. S i., Warsaw, 2007, pp. 303-313. · Zbl 1126.47039
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