This is the last problem.

If A is the algebra of all continuous functions definied on the unit disc, holomorfic in its interior, then the mapping

$\displaystyle T:A \rightarrow A \oplus A$

given by the formula

$\displaystyle x(t) \rightarrow [x(2t),0]$

is a multiplicative mapping satisfying neither

$\displaystyle \sigma(Tx) \in \sigma(x)$

nor

$\displaystyle \sigma(x) \in \sigma(Tx)$.

Once more many thanks for any help or advices.