The trace of one matrix on the image or kernel of another

Question

Suppose $A$ and $B$ are two commuting linear maps $V \to V$. Then $A$ preserves the image and kernel of $B$, and its trace restricted to these subspaces do not depend on a choice of basis. Is there a formula for either of these traces that is basis independent, say as a trace of some polynomial in $A, B, \dots$? Or slightly weaker, is there a way to compute these numbers while choosing only a basis of $V$ (thus making $A$ and $B$ into matrices), but not a basis of $\operatorname{im} B$ or $\ker B$?