  The variable two-step backward differentiation formula (BDF2) is revisited
via a new theoretical framework using the positive semi-definiteness of BDF2
convolution kernels and a class of orthogonal convolution kernels. We prove
that, if the adjacent time-step ratios
$r_k:=\tau_k/\tau_{k-1}\le(3+\sqrt{17})/2\approx3.561$, the adaptive BDF2
time-stepping scheme for linear reaction-diffusion equations is unconditionally
stable and (maybe, first-order) convergent in the $L^2$ norm. The second-order
temporal convergence can be recovered if almost all of time-step ratios $r_k\le
1+\sqrt{2}$ or some high-order starting scheme is used. Specially, for linear
dissipative diffusion problems, the stable BDF2 method preserves both the
energy dissipation law (in the $H^1$ seminorm) and the $L^2$ norm monotonicity
at the discrete levels. An example is included to support our analysis.
