Suppose $U$ is a unitary operator, $A$ a vector self-adjoint operator and $v$ a fixed vector. Is it true that $U[v\cdot (U^\dagger A U)]^2U^\dagger$ equals to $(v\cdot A)^2$? I am mainly confused because of the dotting with $v$. Thank you.
For example, if it is simply $U[(U^\dagger A U)]^2U^\dagger$ then clearly this equals to $A$. but I don't understand how to deal with the $v\cdot$
Added Context: As Muphrid pointed out, "the linear operators act on functions, but the vector v belongs to a finite vector space (and hence, A is a vector of linear operators)".