  In this work the L2-1$_\sigma$ method on general nonuniform meshes is studied
for the subdiffusion equation. When the time step ratio is no less than
$0.475329$, a bilinear form associated with the L2-1$_\sigma$
fractional-derivative operator is proved to be positive semidefinite and a new
global-in-time $H^1$-stability of L2-1$_\sigma$ schemes is then derived under
simple assumptions on the initial condition and the source term. In addition,
the sharp $L^2$-norm convergence is proved under the constraint that the time
step ratio is no less than $0.475329$.
