Constantin Buse, Toka Diagana, Lan Thanh Nguyen, Donal O'Regan
Let T be a strongly continuous semigroup acting on a complex Banach space X and let A be its infinitesimal generator. It is well-known [29,33] that the uniform spectral bound of the semigroup T is negative provided that all solutions to the Cauchy problems
are bounded (uniformly with respect to the parameter . In particular, if X is a Hilbert space, then this yields all trajectories of the semigroup T are exponentially stable, but if X is an arbitrary Banach space this result is no longer valid. Let denote the space of all continuous and 1-periodic functions whose sequence of Fourier-Bohr coefficients belongs to . Endowed with the norm it becomes a non-reflexive Banach space . A subspace of X (related to the pair ) is introduced in the third section of this paper. We prove that the semigroup T is uniformly exponentially stable provided that in addition to the above-mentioned boundedness condition, . An example of a strongly continuous semigroup (which is not uniformly continuous) and fulfills the second assumption above is also provided. Moreover an extension of the above result from semigroups to 1-periodic and strongly continuous evolution families acting in a Banach space is given. We also prove that the evolution semigroup associated with T on does not verify the spectral determined growth condition. More precisely, an example of such a semigroup with uniform spectral bound negative and uniformly growth bound non-negative is given. Finally we prove that the assumption is not needed in the discrete case.
Submitted December 19, 2018. Published June 4, 2019.
Math Subject Classifications: 35B35, 47A30, 46A30.
Key Words: Uniform exponential stability; growth bounds for semigroups; evolution semigroups; exponentially bounded evolution families of operators; Integral equations in Banach spaces; Fourier series.
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| Constantin Buse |
Politehnica University of Timisoara
Department of Mathematics, Piata Victoriei No. 2
| Toka Diagana |
University of Alabama in Huntsville
Department of Mathematical Sciences
301 Sparkman Drive, Shelby Center
Huntsville, AL 35899, USA
| Lan Thanh Nguyen |
Western Kentucky University
Department of Mathematics
Bowling Green, KY 42101, USA
| Donal O'Regan |
National University of Ireland
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