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.

- All seminars.
- Seminars for 2014

Last updated: 06 Nov 2024