Is matrix polynomial of a self adjoint operator necessarily self adjoint?

Question

Let $A$ be a self adjoint matrix on $\Bbb C^n$ and $p(x)$ be a non constant complex polynomial. Then can we say that $p(A)$ is also self adjoint?

I think the above is not necessarily true in general. For instance if I take the polynomial $p(x)=ix$. Then $p(A)=iA$. Then for any $x,y \in \Bbb C^n$ we have $\left <iAx,y \right >=i\left <Ax,y \right >$ whereas $\left <x,iAy \right >=-i\left <Ax,y \right >$. If $p(A)$ was self adjoint we should have $2i\left <Ax,y \right >=0$ for any $x,y \in \Bbb C^n$. In particular if we take $y=Ax$ then $\left <Ax,y \right > > 0$. But then we should have $2i=0$ which can't be true.

Answer 1

Your counterexample is O.K.

It is easy to se that if $A$ is self-adjoint and p is a complex polynomial, then $p(A)$ is normal.