I have the following question. Let $T\colon \mathcal{H} \to \mathcal{H}$ be a bounded, self-adjoint operator on a Hilbert-space $\mathcal{H}$. By spectral theorem we know that there exists a measure space $(\Omega, \Sigma, \lambda)$ and a bounded measurable function $f : \Omega \rightarrow \mathbb{R}$ and a unitary operator $U \colon \mathcal{H} \to L^{2}(\Omega)$ such that $(UTU^{-1}) \varphi = f \varphi$. My question is: Is the spectrum of $T$ and the spectrum of $UTU^{-1}$ then the same, i.e. $\sigma(T) = \sigma(UTU^{-1})$ ? Hope this question is not too trivial? Thanks in advance.
mika
