#### PhD Thesis defence

##### Venue: Lecture Hall I, Department of Mathematics

This thesis investigates two different types of problems in multi-variable operator theory. The first one deals with the defect sequence for a contractive tuple and the second one deals with wandering subspaces of the Bergman space and the Dirichlet space over the polydisc. These are described in (I) and (II) below.

I. We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. We show that there are upper bounds for the defect dimensions. The upper bounds are different in the non-commutative and in the commutative case. The tuples for which these upper bounds are obtained are called maximal contractive tuples. We show that the creation operator tuple on the full Fock space and the co-ordinate multipliers on the Drury-Arveson space are maximal. We also show that if M is an invariant subspace under the creation operator tuple on the full Fock space, then the restriction is always maximal. But the situation is starkly different for co-invariant subspaces. A characterization for a contractive tuple to be maximal is obtained. We define the notion of maximality for a submodule of the Drury-Arveson module on the d-dimensional unit ball. For $d=1$, it is shown that every submodule of the Hardy module over the unit disc is maximal. But for $d>2$, we prove that any homogeneous submodule or a submodule generated by polynomials is not maximal. We obtain a characterization of maximal submodules of the Drury-Arveson module. We also study pure tuples and see how the defect dimensions play a role in their irreducibility.

II. We investigate the following question : Let $(T_1, ....., T_n)$ be a commuting $n$-tuple of bounded linear operators on a Hilbert space $H$. Does there exist a generating wandering subspace $W$ for $(T_1, ....., T_n)$? We got some affirmative answers for the doubly commuting invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc. We show that for any doubly commuting invariant subspace of the Bergman space or the Dirichlet space over polydisc, the tuple consisting of restrictions of co-ordinate multiplication operators always possesses a generating wandering subspace.

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Last updated: 06 Mar 2020