Colloquium: Rank of a shift invariant subspace of the Hardy space H^2 over the bidisc.
Title: Rank of a shift invariant subspace of the Hardy space H2 over the bidisc.
The rank of a bounded linear operator on a Hilbert space is an important numerical invariant. Very briefly, the rank of a bounded linear operator is the cardinality of a minimal generating set. One of the most intriguing and important open problems in operator theory and function theory, closely related to the invariant subspace problem, is the existence of non-trivial generating set for a commuting tuple of operator. Also one may ask when the rank of a commuting tuple of operators is finite. In connection with that, a particular version of the celebrated invariant subspace theorem of Beurling says that a shift invariant (or, shift co-invariant) subspace of the one variable Hardy space H2 is of rank one.
In this talk, I will introduce the notion of a rank corresponding to a single operator and then to an n-tuple of commuting bounded operators on an arbitrary Hilbert space. Then I will explain its connection with a shift invariant subspace of the Hardy space H2 over the bidisc. Note that the rank of a shift invariant subspace S is same as the rank of the shift operator restricted to S. Next, I will introduce the concept of a co-doubly commutativity of a shift invariant subspace and show that the rank of a non-trivial co-doubly commuting shift invariant subspace is exactly 2. As a consequence of this, I will give an affirmative answer to the conjecture of Douglas and Yang, which says that we can write a rank one co-doubly commuting shift invariant subspace as an inner function of one variable times the full Hardy space H2.
Contact Name: Anna Skripka