Semigroup Growth Bounds

B: Semigroup growth bounds

L∞-estimates for the torsion function and L∞-growth of semigroups satisfying Gaussian bounds

We investigate selfadjoint C0-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the long time behaviour of the L∞ operator norm of the semigroup. As an application we prove a new L∞-bound for the torsion function of a Euclidean domain that is cl...

On the Minimal Length of Semigroup Presentations

We define the length of a semigroup presentation and related ideas. Theoretical results and bounds are presented, gained while investigating the lengths of groups defined by semigroup presentations. Minimal length semigroup presentations are obtained in certain cases. AMS Mathematics Subject Classification (2000): 20F05, 20M05

The Growth Bound for Strongly Continuous Semigroups on Fréchet Spaces

We introduce the concepts of growth and spectral bound for strongly continuous semigroups acting on Fréchet spaces and show that the Banach space inequality s(A) 6 ω0(T ) extends to the new setting. Via a concrete example of an even uniformly continuous semigroup we illustrate that for Fréchet spaces effects with respect to these bounds may happen that cannot occur on a Banach space.

Exponential Decay of Semigroups in Hilbert Space

We give simple bounds for the exponential decay of a strongly continuous semigroup V (t) = exp(At) in a Hilbert space. The bounds are expressed by means of the solution X of the Ljapunov equation A X + XA = ?I and the quantity sup 0th kV (t)k for small h. Our main bound is attained on any normal semigroup and can therefore not be improved in general. Let V (t) be a strongly continuous semigroup...