This is an announcement for the paper "Functional calculus for $C_{0}$-groups using (co)type" by Jan Rozendaal.
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 $-\mathrm{i}A$ 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 $\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)$ to $X$, i.e.\ $f(A):\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)\to X$ is bounded for each bounded holomorphic function $f$ on a sufficiently large strip. %Hence $A$ has a bounded calculus for the class of bounded holomorphic functions which decay polynomially of order $\alpha>\frac{1}{p}-\frac{1}{q}$ at infinity. Under additional geometric assumptions, satisfied by $\mathrm{L}^{p}$-spaces, we cover the case $\alpha=\frac{1}{p}-\frac{1}{q}$. 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.
Archive classification: math.FA math.NA
Mathematics Subject Classification: Primary 47A60, Secondary 47D03, 46B20, 42A45
Remarks: 25 pages
Submitted from: janrozendaalmath@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1508.02036
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