On Sylvester operator equations, complete trajectories, regular admissibility, and stability of C0-semigroups
| dc.contributor.author | Immonen, Eero | |
| dc.date.accessioned | 2021-05-28T17:08:58Z | |
| dc.date.available | 2021-05-28T17:08:58Z | |
| dc.date.issued | 2005-06-30 | |
| dc.description.abstract | We show that the existence of a nontrivial bounded uniformly continuous (BUC) complete trajectory for a C0-semigroup TA(t) generated by an operator A in a Banach space X is equivalent to the existence of a solution Π = δ0 to the homogeneous operator equation ΠS|M = AΠ. Here S|M generates the shift C0-group TS(t)|M in a closed translation-invariant subspace M of BUC (ℝ, X), and δ0 is the point evaluation at the origin. If, in addition, M is operator-invariant and 0 ≠ Π ∈ L(M, X) is any solution of ΠS|M = AΠ, then all functions t → ΠTs(t)|Mƒ, ƒ ∈ M, are complete trajectories for TA(t) in M. We connect these results to the study of regular admissibility of Banach function spaces for TA(t); among the new results are perturbation theorems for regular admissibility and complete trajectories. Finally, we show how strong stability of a C0-semigroup can be characterized by the nonexistence of non-trivial bounded complete trajectories for the sun-dual semigroup, and by the surjective solvability of an operator equation ΠS|M = AΠ. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 14 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | Immonen, E. (2005). On Sylvester operator equations, complete trajectories, regular admissibility, and stability of C0-semigroups. <i>Electronic Journal of Differential Equations, 2005</i>(71), pp. 1-14. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/13672 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University-San Marcos, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2005, San Marcos, Texas: Texas State University-San Marcos and University of North Texas. | |
| dc.subject | Sylvester operator equation | |
| dc.subject | Regularly admissible space | |
| dc.subject | Complete nontrivial trajectory | |
| dc.subject | C0-semigroup | |
| dc.subject | Exponential stability | |
| dc.subject | Strong stability | |
| dc.subject | Exponential dichotomy | |
| dc.title | On Sylvester operator equations, complete trajectories, regular admissibility, and stability of C0-semigroups | |
| dc.type | Article |