Corollary 2

Let $p$ be an integer $\geqslant$ 1. For every family $(f_1, . . . , f_p)$ of elements of $\mathscr{O}$(Sp$^n(\boldsymbol{a})$), one has

Sp$^p((\Theta_{\boldsymbol{a}}(f_1), . . . ,\Theta_{\boldsymbol{a}}(f_p))) = (f_1, . . . , f_p$)(Sp$^n(\boldsymbol{a})$).

In particular, for every $f\in \mathscr{O}$(Sp$^n(\boldsymbol{a})$), one has Sp(Θ$_{\boldsymbol{a}}(f)$) $=f$(Sp$^n(\boldsymbol{a})$).

This follows from Corollary 1 and the definition of the joint spectrum.