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I need help with this task because I dont fully understand what unitarily similar operators are.

A is normal operator on Hilbert space H. If A^2 is self-adjoint and dim H < ∞, prove that A is unitarily similar to an operator of the form B⊕C where B∗ = B and C∗ = −C.

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  • $\begingroup$ An operator $A$ is said to be unitarily-equivalent to an operator $B$ if there exists a unitary operator $U$ such that $A=UBU^*$. $\endgroup$ – Chee Han Jun 3 '17 at 14:41
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Just to be clear about conventions, I take my inner products to be linear in the second variable and conjugate-linear in the first.

First, since $H$ is finite dimensional and $A$ is normal, by the (finite-dimensional) spectral theorem, there exist complex numbers $\{\lambda_1,\dotsc,\lambda_{\dim H}\}$ and an orthonormal basis $\{h_1,\dotsc,h_{\dim H}\}$ such that $$ \forall 1 \leq k \leq \dim H, \quad Ah_k = \lambda_k h_k; $$ in other words, $\{h_1,\dotsc,h_{\dim H}\}$ is an ordered orthonormal basis for $H$ consisting of eigenvectors of $A$ with corresponding eigenvalues $\{\lambda_1,\dotsc,\lambda_{\dim H}\}$ of $A$. In particular, if $U_0 : H \to \mathbb{C}^{\dim H}$ is the unitary defined by $$ \forall h \in H, \quad U_0h := \begin{pmatrix} \langle h_1, h \rangle \\ \vdots \\ \langle h_{\dim H}, h \rangle \end{pmatrix}, $$ then $$ \forall v = \begin{pmatrix}v_1 \\ \vdots \\ v_{\dim H}\end{pmatrix} \in \mathbb{C}^{\dim H}, \quad U_0 A U_0^\ast v = \begin{pmatrix}\lambda_1 v_1 \\ \vdots \\ \lambda_{\dim H} v_{\dim H}\end{pmatrix}, $$ so what you have is precisely a "unitary diagonalisation" of $A$.

Next, a unitary diagonalisation of $A$ automatically yields a unitary diagonalisation of $A^2$, since $$ \forall 1 \leq k \leq \dim H, \quad A^2 h_k = \lambda_k^2 h_k, $$ so that $\{h_1,\dotsc,h_{\dim H}\}$ is an ordered orthonormal basis for $H$ consisting of eigenvectors of $A^2$ with corresponding eigenvalues $\{\lambda_1^2,\dotsc,\lambda_{\dim H}^2\}$ of $A^2$. But now,

  1. What do you know about the eigenvalues of a self-adjoint operator?
  2. Given what you know about the eigenvalues $\{\lambda_1^2,\dotsc,\lambda_{\dim H}^2\}$ of the self-adjoint operator $A^2$, what can you conclude about the eigenvalues $\{\lambda_1,\dotsc,\lambda_{\dim H}\}$ of the normal operator $A$?
  3. Can you now modify $U_0$ to a unitary $U : H \to \mathbb{C}^{\dim H}$ such that $U A U^\ast = B \oplus C$ for $B$ self-adjoint and $C$ skew-adjoint? This will come from splitting the eigenvalues $\{\lambda_1,\dotsc,\lambda_{\dim H}\}$ of $A$ into two subsets (by step 2) and then reordering the ordered orthonormal basis $\{h_1,\dotsc,h_{\dim H}\}$ accordingly.
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