# Prove or disprove the following statement on the spectrum of $p(A,A^*).$

Let $$A$$ be a compact linear operator on a Hilbert space $$\mathcal H.$$ Let $$p(A,A^*) = \sum\limits_{i,j = 1}^{k} a_{ij} A^i {A^*}^j$$ be a polynomial in $$A$$ and $$A^*.$$ Here $$a_{ij} \in \mathbb C$$ for all $$i,j = 1,2, \cdots, k.$$ Prove or disprove

$$\sigma (p(A,A^*)) = \left \{p(\lambda, \overline {\lambda})\ \big |\ \lambda \in \sigma (A) \right \}.$$

I can able to show that $$\left \{p(\lambda, \overline {\lambda})\ \big |\ \lambda \in \sigma (A) \right \} \subseteq \sigma (p(A,A^*)).$$ But I don't think that the other part of the inclusion holds true. But I couldn't able to find a suitable counter-example. Could anyone please help me in this regard?

Thanks a bunch.

Let $$p(X,Y) = -X+Y$$, $$\mathcal H = \mathbb C^2$$ and $$A= \begin{pmatrix}0&1\\0&0\end{pmatrix}$$. Then, we have: $$\sigma(A) = \sigma(A^*)= \{0\} \qquad \text{and} \qquad \{p(\lambda,\bar\lambda):\lambda\in\sigma(A)\} = \{0\}$$ while : $$p(A,A^*) = \begin{pmatrix} 0&-1\\ 1&0\end{pmatrix}$$ and : $$\sigma(p(A,A^*)) = \{i,-i\}$$