  We study the vector-valued spectrum $\mathcal{M}_\infty(B_{c_0},B_{c_0})$,
that is, the set of non null algebra homomorphisms from $\mathcal
H^\infty(B_{c_0})$ to $\mathcal H^\infty(B_{c_0})$ which is naturally projected
onto the closed unit ball of $\mathcal H^\infty(B_{c_0}, \ell_\infty)$,
likewise the scalar-valued spectrum $\mathcal M_\infty(B_{c_0})$ which is
projected over $\bar{B}_{\ell_\infty}$. Our itinerary begins in the
scalar-valued spectrum $\mathcal{M}_\infty(B_{c_0})$: by expanding a result by
Cole, Gamelin and Johnson (1992) we prove that on each fiber there are $2^c$
disjoint analytic Gleason isometric copies of $B_{\ell_\infty}$. For the
vector-valued case, building on the previous result we obtain $2^c$ disjoint
analytic Gleason isometric copies of
$B_{\mathcal{H}^\infty(B_{c_0},\ell_\infty)}$ on each fiber. We also take a
look at the relationship between fibers and Gleason parts for both
vector-valued spectra $\mathcal{M}_{u,\infty}(B_{c_0},B_{c_0})$ and
$\mathcal{M}_\infty(B_{c_0},B_{c_0})$.

Functional Analysis;Complex Variables