Abstract
Let
$T = (T_1, \ldots , T_n)$
be a commuting tuple of bounded linear operators on a Hilbert space
$\mathcal{H}$
. The multiplicity of
$T$
is the cardinality of a minimal generating set with respect to
$T$
. In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let
$n \geq 2$
, and let
$\mathcal{Q}_i$
,
$i = 1, \ldots , n$
, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in
$\mathbb {C}$
. If
$\mathcal{Q}_i^{\bot }$
,
$i = 1, \ldots , n$
, is a zero-based shift invariant subspace, then the multiplicity of the joint
$M_{\textbf {z}} = (M_{z_1}, \ldots , M_{z_n})$
-invariant subspace
$(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{\perp }$
of the Dirichlet space or the Hardy space over the unit polydisc in
$\mathbb {C}^{n}$
is given by
\[ \mbox{mult}_{M_{\textbf{{z}}}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}}}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}} = \sum_{i=1}^{n} (\mbox{mult}_{M_z|_{\mathcal{Q}_i^{{\perp}}}} (\mathcal{Q}_i^{\bot})) = n. \]
A similar result holds for the Bergman space over the unit polydisc.
Invariant subspaces of the harmonic Dirichlet space with large co-dimension