Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rank of a co-doubly commuting submodule is 2   We prove that the rank of a non-trivial co-doubly commuting submodule is $2$.
More precisely, let $\varphi, \psi \in H^\infty(\mathbb{D})$ be two inner
functions. If $\mathcal{Q}_{\varphi} = H^2(\mathbb{D})/ \varphi
H^2(\mathbb{D})$ and $\mathcal{Q}_{\psi} = H^2(\mathbb{D})/ \psi
H^2(\mathbb{D})$, then \[ \mbox{rank~}(\mathcal{Q}_{\varphi} \otimes
\mathcal{Q}_{\psi})^\perp = 2. \] An immediate consequence is the following:
Let $\mathcal{S}$ be a co-doubly commuting submodule of $H^2(\mathbb{D}^2)$.
Then $\mbox{rank~} \mathcal{S} = 1$ if and only if $\mathcal{S} = \Phi
H^2(\mathbb{D}^2)$ for some one variable inner function $\Phi \in
H^\infty(\mathbb{D}^2)$. This answers a question posed by R. G. Douglas and R.
Yang.

37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics