Spectrum in an separable Hilbert space

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 $\sigma(T) = \overline{\{\lambda_i\}}$.

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