Operator with a single point as spectrum

Question

Give an example of a bounded operator acting on a Hilbert space, whose spectrum consists of a single point. I can only think of the $0$ operator an a multiple of the identity, is there any other simple but non-trivial example? Thanks

Answer 1

Hint:

$$ M=\begin{pmatrix} a&1\\0&a \end{pmatrix} $$

has the only eigenvalue $a$. For an $n\times n$ matrix you can find more examples using different Jordan blocks with the same diagonal elements.

Answer 2

Let $A$ be an operator on an Hilbert $H$ space such that $A$ has empty spectrum. Define $$\begin{array}{rccc}A^\star\colon&H\oplus\mathbb{C}&\longrightarrow&H\oplus\mathbb{C}\\&(v,z)&\mapsto&\bigl(A(v),0\bigr).\end{array}$$Then the spectrum of $A^\star$ is $\{0\}$, but $A^\star$ is not a multiple of the identity.