Bounded $H^{\infty }$-calculus for vectorial-valued operators with Gaussian kernel estimates

Abstract

We prove that the vector-valued generator of a bounded holomorphic semigroup represented by a kernel satisfying Gaussian estimates with bounded $H^{\infty }$-calculus in $L^2({\mathbb{R}}^d;{\mathbb{C}}^m)$ admits bounded $H^{\infty }$-calculus for every $p\in (1,\infty )$. We apply this result to the elliptic operator $-\mathrm{div}(Q\nabla )+V$, where the potential term V is a matrix-valued function whose entries belong to $L_{\mathrm{loc}}^1({\mathbb{R}}^d)$ and, for almost every $x\in {\mathbb{R}}^d$, $V(x)$ is a symmetric and nonnegative definite matrix.