# Journal of Operator Theory

Volume 82, Issue 2, Fall 2019 pp. 445-468.

Characterization of invariant subspaces in the polydisc

**Authors**:
Amit Maji (1), Aneesh Mundayadan (2), Jaydeb Sarkar
(3), T.R. Sankar (4)

**Author institution:**(1) Indian Statistical Institute, Statistics and
Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

(2) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile,
Mysore Road, Bangalore, 560059, India

(3) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile,
Mysore Road, Bangalore, 560059, India

(4) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

**Summary: **We give a complete characterization of invariant subspaces for
$(M_{z_1}, \ldots, M_{z_n})$ on the Hardy space $H^2(\mathbb{D}^n)$
over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$, $n >1$. In
particular, this yields a complete set of unitary invariants for
invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on
$H^2(\mathbb{D}^n)$. As a consequence, we classify a large class of
$n$-tuples of commuting isometries. All of our results hold for
vector-valued Hardy spaces over $\mathbb{D}^n$, $n
> 1$.

**DOI: **http://dx.doi.org/10.7900/jot.2018jun07.2204

**Keywords: **invariant subspaces, commuting isometries, Hardy space,
polydisc, bounded analytic functions

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