We obtain polynomial decay rates for C₀-semigroups, assuming that the resolvent grows polynomially at infinity in the complex right half-plane. Our results do not require the semigroup to be uniformly bounded, and for unbounded semigroups, we improve upon previous results by, for example, removing a logarithmic loss on non-Hilbertian Banach spaces.

We study growth rates for strongly continuous semigroups. We prove that a growth rate for the resolvent on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroup is positive and the underlying space is an (Formula presented.)-space or a space of continuous functions. We also prove variations of the main results on fractional domains; these are valid on more general Banach spaces. In the second part of the article, we apply our main theorem to prove optimality in a classical example by Renardy of a perturbed wave equation which exhibits unusual spectral behavior.

We study polynomial and exponential stability for C₀-semigroups using the recently developed theory of operator-valued (L^p,L^q) Fourier multipliers. We characterize polynomial decay of orbits of a C₀-semigroup in terms of the (L^p,L^q) Fourier multiplier properties of its resolvent. Using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space. We do not assume that the semigroup is uniformly bounded, our results depend only on spectral properties of the generator. As a corollary of our work on polynomial stability we reprove and unify various existing results on exponential stability, and we also obtain a new theorem on exponential stability for positive semigroups.

In this article, we consider Fourier multiplier operators between vector-valued Besov spaces with different integrability exponents p and q, which depend on the type p and cotype q of the underlying Banach spaces. In a previous article, we considered L^p-L^q multiplier theorems. In the current article, we show that in the Besov scale one can obtain results with optimal integrability exponents. Moreover, we derive a sharp result in the L^p-L^q setting as well. We consider operator-valued multipliers without smoothness assumptions. The results are based on a Fourier multiplier theorem for functions with compact Fourier support. If the multiplier has smoothness properties, then the boundedness of the multiplier operator extrapolates to other values of p and q for which 1/p - 1/q remains constant.

In this paper we develop the theory of Fourier multiplier operators (Formula presented.), for Banach spaces X and Y, (Formula presented.) and (Formula presented.) an operator-valued symbol. The case (Formula presented.) has been studied extensively since the 1980s, but far less is known for (Formula presented.). In the scalar setting one can deduce results for (Formula presented.) from the case (Formula presented.). However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that X and Y are UMD spaces and that m satisfies a smoothness condition. We show that for (Formula presented.) other geometric conditions on X and Y, such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for (Formula presented.) without any smoothness properties of m. Under smoothness conditions the boundedness results can be extrapolated to other values of p and q as long as (Formula presented.) remains constant.

Let X, Y be Banach spaces and let L(X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on L(X,Y) and apply this theory to obtain commutator estimates of the form

We also study the estimate above in the setting of Banach ideals in L(X,Y). The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix.