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Sharp polynomial bounds for certain \(C_0\)-groups generated by operators with non-basis family of eigenvectors. (English) Zbl 07306995

Summary: Sharp polynomial bounds for norms of \(C_0\)-groups generated by operators with purely imaginary eigenvalues \(\lambda_n = i \ln n, n \in \mathbb{N}\), and complete minimal non-basis family of eigenvectors, constructed recently by G. Sklyar and V. Marchenko in [17], are obtained. Besides, it is shown that these \(C_0\)-groups do not have a maximal asymptotics. For the more general case of behaviour of the spectrum of operators we present the proof of one lemma concerning the behaviour of \(j\)-th differences of sequences of complex exponentials \(\exp(itf(n)), n \in \mathbb{N}\), that is used in [17] to obtain bounds from above for norms of corresponding \(C_0\)-groups. Also the growth properties of the resolvent for generators of constructed \(C_0\)-groups are discussed.

MSC:

47D06 One-parameter semigroups and linear evolution equations
34G10 Linear differential equations in abstract spaces
46B45 Banach sequence spaces
34K25 Asymptotic theory of functional-differential equations

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[1] Amrein, W. O.; de Monvel, A. Boutet; Georgescu, V., \( C_0\)-Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians, Modern Birkhäuser Classics (1996), Birkhäuser: Birkhäuser Basel · Zbl 0962.47500
[2] Arendt, W.; Grabosch, A.; Greiner, G.; Groh, U.; Lotz, H. P.; Moustakas, U.; Nagel, R.; Neubrander, F.; Schlotterbeck, U.; Nagel, Rainer, One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, vol. 1184 (1986), Springer-Verlag: Springer-Verlag Berlin
[3] Barnes, B. A., Operators which satisfy polynomial growth conditions, Pac. J. Math., 138, 2, 209-219 (1989) · Zbl 0693.47001
[4] Davies, E. B., Non-self-adjoint differential operators, Bull. Lond. Math. Soc., 34, 5, 513-532 (2002) · Zbl 1052.47042
[5] Davies, E. B., Linear Operators and Their Spectra, Cambridge Studies in Advanced Mathematics, vol. 106 (2007), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1138.47001
[6] Dunford, N.; Schwartz, J. T., Linear Operators, Part 1: General Theory (1958), Interscience Publishers: Interscience Publishers New York, London, MR 117523
[7] Eisner, T., Polynomially bounded \(C_0\)-semigroups, Semigroup Forum, 70, 1, 118-126 (2005) · Zbl 1102.47027
[8] Eisner, T., Stability of Operators and Operator Semigroups, Oper. Theory Adv. Appl., vol. 209 (2010), Birkhäuser: Birkhäuser Basel · Zbl 1205.47002
[9] Eisner, T.; Zwart, H., A note on polynomially growing \(C_0\)-semigroups, Semigroup Forum, 75, 2, 438-445 (2007) · Zbl 1135.47043
[10] Engel, K.-J.; Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194 (2000), Springer-Verlag: Springer-Verlag New York · Zbl 0952.47036
[11] Goldstein, J. A.; Wacker, M., The energy space and norm growth for abstract wave equations, Appl. Math. Lett., 16, 767-772 (2003) · Zbl 1043.35121
[12] Lyubich, Y. I.; Phong, V. Q., Asymptotic stability of linear differential equation in Banach space, Stud. Math., 88, 37-42 (1988) · Zbl 0639.34050
[13] Malejki, M., \( C_0\)-groups with polynomial growth, Semigroup Forum, 63, 3, 305-320 (2001) · Zbl 1034.47014
[14] Miloslavskii, A. I., Stability of certain classes of evolution equations, Sib. Math. J., 26, 5, 723-735 (1985) · Zbl 0659.35050
[15] Sklyar, G. M., On the maximal asymptotics for linear differential equations in Banach spaces, Taiwan. J. Math., 14, 2203-2217 (2010) · Zbl 1230.34050
[16] Sklyar, G. M.; Marchenko, V. A., Hardy inequality and the construction of the generator of a \(C_0\)-group with eigenvectors not forming a basis, Dopov. Nats. Akad. Nauk Ukr., 9, 13-17 (2015) · Zbl 1340.47087
[17] Sklyar, G. M.; Marchenko, V., Hardy inequality and the construction of infinitesimal operators with non-basis family of eigenvectors, J. Funct. Anal., 272, 3, 1017-1043 (2017) · Zbl 1359.47042
[18] Sklyar, G. M.; Marchenko, V., Resolvent of the generator of the \(C_0\)-group with non-basis family of eigenvectors and sharpness of the XYZ theorem, J. Spectr. Theory (2018), accepted
[19] Sklyar, G. M.; Polak, P., Asymptotic growth of solutions of neutral type systems, Appl. Math. Optim., 67, 3, 453-477 (2013) · Zbl 1282.34077
[20] Sklyar, G. M.; Polak, P., On asymptotic estimation of a discrete type \(C_0\)-semigroups on dense sets: application to neutral type systems, Appl. Math. Optim., 75, 2, 175-192 (2017) · Zbl 1377.34100
[21] Sklyar, G. M.; Polak, P., Notes on the asymptotic properties of some class of unbounded strongly continuous semigroups, J. Math. Phys. Anal. Geom., 15, 3, 412-424 (2019) · Zbl 1461.34080
[22] Sklyar, G. M.; Shirman, V., On asymptotic stability of linear differential equation in Banach space, Teor. Funkc. Funkc. Anal. Ih Prilozh., 37, 127-132 (1982) · Zbl 0521.34063
[23] Trefethen, L. N.; Embree, M., Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators (2005), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1085.15009
[24] Xu, G. Q.; Yung, S. P., The expansion of a semigroup and a Riesz basis criterion, J. Differ. Equ., 210, 1-24 (2005) · Zbl 1131.47042
[25] Zwart, H., Riesz basis for strongly continuous groups, J. Differ. Equ., 249, 2397-2408 (2010) · Zbl 1203.47020
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