We give several conditions implying that the spectral bound of the generator of a -semigroup is negative. Applications to stability theory are considered.
@article{bwmeta1.element.bwnjournal-article-smv125i1p23bwm, author = {Yu. Tomilov}, title = {On the spectral bound of the generator of a $C\_0$-semigroup}, journal = {Studia Mathematica}, volume = {122}, year = {1997}, pages = {23-33}, zbl = {0892.47046}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv125i1p23bwm} }
Tomilov, Yu. On the spectral bound of the generator of a $C_0$-semigroup. Studia Mathematica, Tome 122 (1997) pp. 23-33. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv125i1p23bwm/
[00000] [1] C. J. K. Batty, Tauberian theorems for the Laplace-Stieltjes transform, Trans. Amer. Math. Soc. 322 (1990), 783-804. | Zbl 0716.44001
[00001] [2] C. J. K. Batty, Asymptotic behaviour of semigroups of operators, in: Functional Analysis and Operator Theory, J. Zemánek (ed.), Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., Warszawa, 1994, 35-52. | Zbl 0818.47034
[00002] [3] P. Clément et al., One-Parameter Semigroups, CWI Monograph 5, North-Holland, 1987.
[00003] [4] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980. | Zbl 0457.47030
[00004] [5] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Colloq. Publ. 31, Amer. Math. Soc., Providence, R.I., 1957. | Zbl 0078.10004
[00005] [6] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N.J., 1962. | Zbl 0117.34001
[00006] [7] A. Lebow, Spectral radius of an absolutely continuous operator, Proc. Amer. Math. Soc. 36 (1972), 511-514. | Zbl 0273.47001
[00007] [8] G. Mil'shteĭn, Extension of semigroups to locally convex spaces, Izv. Vuz. Mat. 2 (1977), 91-95 (in Russian).
[00008] [9] W. Mlak, On a theorem of Lebow, Ann. Polon. Math. 35 (1977), 107-109. | Zbl 0371.47007
[00009] [10] J. van Neerven, The Asymptotic Behavior of Semigroups of Linear Operators, Oper. Theory Adv. Appl. 88, Birkhäuser, Basel, 1996.
[00010] [11] N. K. Nikol'skiĭ, A Tauberian theorem on the spectral radius, Siberian Math. J. 18 (1977), 1367-1372.
[00011] [12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983. | Zbl 0516.47023
[00012] [13] M. Slemrod, Asymptotic behavior of -semigroups as determined by the spectrum of the generator, Indiana Univ. Math. J. 25 (1976), 783-792. | Zbl 0313.47026
[00013] [14] G. Weiss, Weak -stability of a linear semigroup on a Hilbert space implies exponential stability, J. Differential Equations 76 (1988), 269-285. | Zbl 0675.47031
[00014] [15] G. Weiss, Weakly -stable operators are power stable, Internat. J. Systems Sci. 20 (1989), 2323-2328. | Zbl 0686.93081
[00015] [16] G. Weiss, The resolvent growth assumption for semigroups in Hilbert space, J. Math. Anal. Appl. 145 (1990), 154-171. | Zbl 0693.47034
[00016] [17] V. Wrobel, Stability and spectra of -semigroups, Math. Ann. 285 (1989), 201-219. | Zbl 0698.47023
[00017] [18] P. Yao and P. Feng, A characteristic condition for the exponential stability of -semigroups, Chinese Sci. Bull. 39 (1994), 534-537. | Zbl 0815.47056
[00018] [19] J. Zabczyk, A note on -semigroups, Bull. Acad. Polon. Sci. Sér. Sci. Math. 23 (1975), 895-898. | Zbl 0312.47037