  We investigate the transverse polarization of the $\Lambda$ hyperon in the
processes $e^+e^-\to\Lambda^\uparrow \pi^\pm X$ and $e^+e^-\to\Lambda^\uparrow
K^\pm X$ within the framework of the transverse momentum dependent~(TMD)
factorization. The transverse polarization is contributed by the convolution of
the transversely polarizing fragmentation function~(PFF) $D_{1T}^\perp$ of the
lambda hyperon and the unpolarized fragmentation function $D_1$ of pion/kaon.
We adopt the spectator diquark model result for $D_{1T}^{\perp}$ to numerically
estimate the transverse polarization in $e^+e^-\to\Lambda^\uparrow h X$ process
at the kinematical region of Belle Collaboration. To implement the TMD
evolution formalism of the fragmentation functions, we apply two different
parametrizations on the nonperturbative Sudakov form factors associated with
the fragmentation functions of the $\Lambda$, pion and kaon. It is found that
our prediction on the polarization in the $\Lambda \pi^+$ production and
$\bar{\Lambda} \pi^-$ is consistent with the recent Belle measurement in size
and sign, while the model predictions on the polarizations in $\Lambda\pi^-$
and $\Lambda K^\pm$ productions show strong disagreement with the Belle data.
The reason for the discrepancies is discussed and possible approaches to
improve the calculation in the future are also discussed.
