Electron. J. Diff. Eqns., Vol. 2005(2005), No. 71, pp. 1-14.
### On Sylvester operator equations, complete trajectories,
regular admissibility, and stability of
-semigroups

Eero Immonen

**Abstract:**

We show that the existence of a nontrivial bounded uniformly
continuous (BUC) complete trajectory for a
-semigroup
generated by an operator
in a Banach space
is equivalent
to the existence of a solution
to the homogenous
operator equation
. Here
generates the shift
-group
in a closed
translation-invariant subspace
of
,
and
is the point evaluation at the origin. If, in addition,
is operator-invariant and
is any solution of
, then all functions
,
,
are complete trajectories for
in
.
We connect these results
to the study of regular admissibility of Banach function spaces for
;
among the new results are perturbation theorems for regular
admissibility and complete trajectories.
Finally, we show how strong stability of a
-semigroup
can be
characterized by the nonexistence of nontrivial bounded complete
trajectories for the sun-dual semigroup, and by the surjective solvability
of an operator equation
.
Submitted April 17, 2005. Published June 30, 2005.

Math Subject Classifications: 47D03.

Key Words: Sylvester operator equation; regularly admissible space;
complete nontrivial trajectory;
-semigroup; exponential stability;
strong stability; exponential dichotomy.

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Eero Immonen
Institute of Mathematics
Tampere University of Technology
PL 553, 33101 Tampere, Finland
email: Eero.Immonen@tut.fi |

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