The partial transpose of an operator $M$ with respect to subsystem $A$ is given by $$ M^{T_A} := \left(\sum_{abcd} M^{ab}_{cd} \underbrace{|a\rangle \langle b| }_{A}\otimes \underbrace{|c \rangle \langle d|}_B\right)= \left(\sum_{abcd} M^{ab}_{cd} |b\rangle \langle a| \otimes |c \rangle \langle d|\right). $$ Is there an identity for the partial transof the product of operators,i.e. $$ (AB)^{T_A}= \quad ? $$ I suspect that in general $(AB)^{T_A} \neq B^{T_A} A^{T_A} $ or even $\neq A^{T_A}B^{T_A}$.

What if $A=B$?

The partial transpose of an operator $M$ with respect to subsystem $A$ is given by $$ M^{T_A} := \left(\sum_{abcd} M^{ab}_{cd} \underbrace{|a\rangle \langle b| }_{A}\otimes \underbrace{|c \rangle \langle d|}_B\right)= \left(\sum_{abcd} M^{ab}_{cd} |b\rangle \langle a| \otimes |c \rangle \langle d|\right). $$ Is there an identity for the partial transpose of the product of operators,i.e. $$ (AB)^{T_A}= \quad ? $$ I suspect that in general $(AB)^{T_A} \neq B^{T_A} A^{T_A} $ or even $\neq A^{T_A}B^{T_A}$.

What if $A=B$?