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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 $[-1,1]$.

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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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The spectrum is the closure of the set $\{c_n,n\geqslant 1\}$. To see that, take an element $\lambda$ which is not in the closure of this set, and show that $T-\lambda I$ is invertible. To see that, fix $\delta$ such that for all $n$, $|\lambda-c_n|>\delta$. So we can define the operator $T\colon e_n\mapsto \frac 1{\lambda-c_n}e_n$.

Then part b) is easily solvable from that.

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