  We study the canonical heat flow $(\mathsf{H}_t)_{t\geq 0}$ on the cotangent
module $L^2(T^*M)$ over an $\mathrm{RCD}(K,\infty)$ space
$(M,\mathsf{d},\mathfrak{m})$, $K\in\boldsymbol{\mathrm{R}}$. We show
Hess-Schrader-Uhlenbrock's inequality and, if $(M,\mathsf{d},\mathfrak{m})$ is
also an $\mathrm{RCD}^*(K,N)$ space, $N\in(1,\infty)$, Bakry-Ledoux's
inequality for $(\mathsf{H}_t)_{t\geq 0}$ w.r.t. the heat flow
$(\mathsf{P}_t)_{t\geq 0}$ on $L^2(M)$. Variable versions of these estimates
are discussed as well. In conjunction with a study of logarithmic Sobolev
inequalities for $1$-forms, the previous inequalities yield various
$L^p$-properties of $(\mathsf{H}_t)_{t\geq 0}$, $p\in [1,\infty]$.
  Then we establish explicit inclusions between the spectrum of its generator,
the Hodge Laplacian $\smash{\vec{\Delta}}$, of the negative functional
Laplacian $-\Delta$, and of the Schr\"odinger operator $-\Delta+K$. In the
$\mathrm{RCD}^*(K,N)$ case, we prove compactness of $\smash{\vec{\Delta}^{-1}}$
if $M$ is compact, and the independence of the $L^p$-spectrum of
$\smash{\vec{\Delta}}$ on $p \in [1,\infty]$ under a volume growth condition.
  We terminate by giving an appropriate interpretation of a heat kernel for
$(\mathsf{H}_t)_{t\geq 0}$. We show its existence in full generality without
any local compactness or doubling, and derive fundamental estimates and
properties of it.
