Two element x, y in a C^{∗}-algebra A such that sp(xy) ≠ sp(yx).

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Let A = B(l^{2}), x be the unilateral shift operator on l^{2}, defined by T (α_{1}, α_{2}, ...) =

(0, α_{1}, α_{2}, ...), and y = T^{∗}. Then T T^{∗}(α_{1}, α_{2}, …) = T (α_{2}, α_{3}, …) = (0, α_{2}, α_{3}, …)

and T^{∗}T (α_{1}, α_{2}, …) = T^{∗}(0, α_{1}, α_{2}, …) = (α_{1}, α_{2}, …). Hence sp(T^{∗}T) = {1} but

0 ∈ sp(T T^{∗}) (since (T T^{∗})(1, 0, 0, …) = (0, 0, …)).