$ \omega $-left approximation dimensions under stable equivalences of adjoint type

In this paper, we study invariant properties of $ \omega $-left approximation dimensions of modules under the stable equivalences of adjoint type. As applications, we prove that Wakamatsu tilting modules, Wakamatsu tilting conjectures, relative $ n $-torsionfree modules, and generalized Gorenstein dimensions of modules are preserved under those equivalences.