| Details: |
In this talk, we discuss various aspects of the well known Invariant Subspace Problem, namely, the problem of deciding if every bounded linear operator on a Banach space possesses a nontrivial invariant subspace. Examples of bounded linear operators on certain Banach spaces for which the only invariant subspaces are the trivial ones have been found. However, the question remains open for a separable Hilbert space. A problem related to the question of the Invariant Subspace Problem is the description of all the invariant subspaces of a fixed operator. As one might expect, this is not an easy task, in general. One exception is the case of the multiplication by the coordinate function on the Hardy space of the unit disc. This is the well known theorem of Beurling. It is an important ingredient in describing the canonical model for a pure contraction. However the natural generalization of Beurling's Theorem for the multiplication by coordinate functions on the Hardy space of the polydisc fails spectacularly. If time permits, we will show how to answer some of the questions that arises due to absence of a theorem like that of Beurling, for a large class of Hilbert modules using techniques from complex geometry, initially developed by Cowen and Douglas. |