Necessary and Sufficient Condition for Eigenspaces be Orthogonal?

Question

Given a linear operator $T$ defined on an inner product space $V$ over $\mathbb C$ or $\mathbb R$. Whether or not the eigenspaces of $T$ are orthogonal subspaces can be determined by the property of $T$, i.e. $T^*T=TT^*$, but is that $TT^*=T^*T$ or $T=T^*$ the necessary and sufficient condition that the eigenspaces be orthogonal subspaces? I mean whether or not there may be other condition(s) that also make the eigenspaces orthogonal?

Answer 1

In case of C ,by spectral theorem of complex vector space, you can show the following are equivalent statement 1) T is normal 2) there exist orthonormal basis of Eigen vectors

Similarly with spectral theorem for real vector space you can show the following are equivalent statement 1) T is Selfadjoint or symmetric 2) there exist orthonormal basis of Eigen vectors