What can we say about eigen values of adjoint of a linear operator

Question

$T: V \to V$ be a linear operator. Adjoint of T is defined as $T^{×}: V' \to V'$ and $T^{×}g = gT$. If $\lambda$ is an eigen value of T such that $T(x)= \lambda x$ for some non zero vector $x\in V$. Then we have $T^{×}g(x)= \lambda g(x)$ . But this doesn't imply $T^{×}g = \lambda g$. Another property of this adjoint operator is that the matrix representation of $T^{×}$ is equal to the transpose of that of $T$. So from this can we say eigen values of $T$ and $T^{×}$ are same.