Exercise 3, § 1

(a) Let $H$ be a Hilbert space admitting a countable orthonormal basis $(e_0, e_1, e_2, . . .)$, and $A =\mathscr{L}(H)$. Let $u\in \mathscr{L}(H)$ such that $u(e_i) =e_{i+1}$ for every $i$. Let $u'\in \mathscr{L}(H)$ such that $u'(e_i) =e_{i-1}$ for $i\geqslant 1$ and $u'(e_0) = 0$. We have $u'u= 1$, but $uu'(e_0) = 0$, so that Sp$_A(u'u)\not=$ Sp$_A(uu')$.

(b) Let A be a unital noetherian algebra over a commutative field. If $x, y\in A$, one has Sp$_A(xy) =$ Sp$_A(yx)$ (use exerc. 6 of A, VIII, p. 37).