# bounded normal operator and spectrum

Problem: If A is a bounded normal operator, the spectrum $$\sigma(A)=\{s+it:s \in \sigma(B),t \in \sigma(C)\}$$, where B, C are bounded self adjoint operators which commute.

Fact: A bounded normal operator A can be written $$A=B+iC$$, where B,C are bounded self adjoint operators which commute.

Fact: Let H be a complex Hilbert space and let $$A:H \rightarrow H$$ be a bounded complex linear operator, then A is normal if only if $$\Vert A^*x \Vert=\Vert Ax \Vert$$ for all $$x \in H$$. Also every self-adjoint operator is normal.

I was told there is a mistake in the problem, but have not spotted it. Thanks in advance.

This is not true even in the simplest of examples. For instance $$\begin{bmatrix} 1&0\\0&i\end{bmatrix} =\begin{bmatrix} 1&0\\0&0\end{bmatrix} +i\begin{bmatrix} 0&0\\0&1\end{bmatrix}.$$ Here $$\sigma(A)=\{1,i\}$$ and $$\sigma(B)=\sigma(C)=\{0,1\}$$. But $$0=0+i0\in\{s+it:\ s\in\sigma(B),\ t\in\sigma(C)\}$$ is not in $$\sigma(A)$$.
What does hold is that $$\sigma(A)\subset \{s+it:\ s\in\sigma(B),\ t\in\sigma(C)\}$$. This can be proven via functional calculus, for instance. If $$f(t)=g(t)+ih(t)$$, then the spectrum of $$M_f$$ is $$\sigma(M_f)=\operatorname{ran}f=\{g(t)+ih(t):\ t\}\subset \{s+it:\ s\in\operatorname{ran}g,\ t\in\operatorname{ran}h\}.$$