Different ways of defining Entanglement

Question

Consider (finite-dim) Hilbert spaces $\mathcal{H},\mathcal{K}$ and let $A$ denote an operator on $\mathcal{H}\otimes \mathcal{K}$. There are then two ways of defining the ``entanglement" of operator $A$.

1. Hilbert-Schmidt operators satisfy $L^2 (\mathcal{H}\otimes \mathcal{K}) \cong L^2 (\mathcal{H})\otimes L^2 (\mathcal{K})$, and thus we can perform Schmidt decomposition on $A$ and define the "entanglement" as the corresponding Schmidt rank

2. Alternatively, each vector $\psi \in \mathcal{H}\otimes \mathcal{K}$ has a Schmidt rank, which we denote $SR(\psi)$. One can also define the ``entanglement" as the supremum of $SR(A\psi)/SR(\psi)$ over all vectors $\psi$.

Is there an equivalenc relation between the two definitions?