An operator T on a Hilbert space such that the set eig(T) of all eigenvalues of T is empty but sp(T) ≠ ∅.

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The unilateral shift operator on the Hilbert space l^{2} ( with its standard orthonormal basis (e_{n})) given by T e_{n} = e_{n+1}, n ∈ N, has no eigenvalue; since obviously 0 ∉ eig(T) and if 0 ≠ λ ∈ eig(T) and T x = λx for some

x = ∑_{n=1}^{∞}α_{n}e_{n} ≠ 0, then ∑_{n=1}^{∞}α_{n}e_{n+1} = ∑_{n=1}^{∞}λα_{n}e_{n} and hence α_{n} = 0 for all n, i.e.

x = 0, a contradiction. Next observe that 0 ∈ sp(T); otherwise T would be invertible so T (T^{−1}(e_{1})) = e_{1} , but < T (T^{−1}e_{1}), e_{1} >= 0 by the definition of T, that is impossible.