  Let $\mathcal{M}(\mathbb{R}^n)$ be the class of bounded away from one and
infinity functions $p:\mathbb{R}^n\to[1,\infty]$ such that the Hardy-Littlewood
maximal operator is bounded on the variable Lebesgue space
$L^{p(\cdot)}(\mathbb{R}^n)$. We show that if $a$ belongs to the H\"ormander
class $S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$, $0\le\delta<1$, then
the pseudodifferential operator $\Op(a)$ is bounded on the variable Lebesgue
space $L^{p(\cdot)}(\R^n)$ provided that $p\in\cM(\R^n)$. Let
$\mathcal{M}^*(\mathbb{R}^n)$ be the class of variable exponents
$p\in\mathcal{M}(\mathbb{R}^n)$ represented as
$1/p(x)=\theta/p_0+(1-\theta)/p_1(x)$ where $p_0\in(1,\infty)$,
$\theta\in(0,1)$, and $p_1\in\mathcal{M}(\mathbb{R}^n)$. We prove that if $a\in
S_{1,0}^0$ slowly oscillates at infinity in the first variable, then the
condition \[ \lim_{R\to\infty}\inf_{|x|+|\xi|\ge R}|a(x,\xi)|>0 \] is
sufficient for the Fredholmness of $\Op(a)$ on $L^{p(\cdot)}(\R^n)$ whenever
$p\in\cM^*(\R^n)$. Both theorems generalize pioneering results by Rabinovich
and Samko \cite{RS08} obtained for globally log-H\"older continuous exponents
$p$, constituting a proper subset of $\mathcal{M}^*(\mathbb{R}^n)$.
