Functional calculus for $C_0$-groups using (co)type

Abstract

We study the functional calculus properties of generators of $C_0$-groups under type and cotype assumptions on the underlying Banach space. In particular, we show the following. Let $-iA$ generate a $C_0$-group on a Banach space $X$ with type $p\in [1,2]$ and cotype $q\in [2,\infty )$. Then $A$ has a bounded ${\mathcal{H}}^{\infty }$-calculus from ${\mathrm{D}}_A(\frac{1}{p}-\frac{1}{q},1)$ to $X$, i.e. $f(A):{\mathrm{D}}_A(\frac{1}{p}-\frac{1}{q},1)\to X$ is bounded for each bounded holomorphic function $f$ on a sufficiently large strip. As a corollary of our main theorem, for sectorial operators we quantify the gap between bounded imaginary powers and a bounded ${\mathcal{H}}^{\infty }$-calculus in terms of the type and cotype of the underlying Banach space. For cosine functions we obtain similar results as for $C_0$-groups. We extend our results to $R$-bounded operator-valued calculi, and we give an application to the theory of rational approximation of $C_0$-groups.

Minor modifications, 27 pages. Published online in Quarterly Journal of Mathematics

References in corpus (2)

• Fourier multiplier theorems involving type and cotype
• Fourier multiplier theorems on Besov spaces under type and cotype conditions

Cited by in corpus (2)

• Fourier multiplier theorems on Besov spaces under type and cotype conditions
• Operator-valued $(L^p,L^q)$ Fourier multipliers and stability theory for evolution equations