Let $H$ be a separable Hilbert space with orthonormal basis $\{e_i\}$. Let $(c_n)$ be a bounded sequence of complex numbers and consider the bounded linear operator $T$ on $H$ defined by $Tx = \sum_{n\geq 1} c_n(x,e_n)e_n$
a) What is the spectrum of $T$?
b) Give an example of a selfadjoint operator on $H$ whose spectrum is $[-1,1]$.
Looking at $(T - \lambda I)x$ where $x = \sum_{n\geq 1} (x,e_n)e_n$ by Parsevall we get the eigenvalues $\lambda_i = c_i$ one guess is that the spectrum $\sigma(T) = \overline{\{\lambda_i\}}$.