Two element a, b of a Banach algebra such that neither r(ab) ≤ r(a)r(b) nor r(a + b) ≤ r(a)r(b).

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Consider a = [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] and b = [ \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ] in M_{2}(C) ≃ B(C^{2}). Then

sp(a) = {0}, sp(b) = {0}, sp(a + b) = {−1, 1}, sp(ab) = {0, 1} and we have required inequalities.