  In this paper, Hunt's hypothesis (H) and Getoor's conjecture for L\'{e}vy
processes are revisited. Let $X$ be a L\'{e}vy process on $\mathbf{R}^n$ with
L\'{e}vy-Khintchine exponent $(a,A,\mu)$. {First, we show that if $A$ is
non-degenerate then $X$ satisfies (H). Second, under the assumption that
$\mu({\mathbf{R}^n\backslash \sqrt{A}\mathbf{R}^n})<\infty$, we show that $X$
satisfies (H) if and only if the equation $$ \sqrt{A}y=-a-\int_{\{x\in
{\mathbf{R}^n\backslash \sqrt{A}\mathbf{R}^n}:\,|x|<1\}}x\mu(dx),\ y\in
\mathbf{R}^n, $$ has at least one solution. Finally, we show that if $X$ is a
subordinator and satisfies (H) then its drift coefficient must be 0.}
