Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators

In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invariant subspaces with a finite defect. This is a powerful result which allows us to gain insight in to the structure of scalar subspaces of the Hardy space using vector-valued Hardy space techniques. These results have far reaching applications, in particular they allow us to develop an all encompassing approach to the study of the kernels of: the Toeplitz operator, the truncated Toeplitz operator, the truncated Toeplitz operator on the multiband space and the dual truncated Toeplitz operator.

Compressions of Multiplication Operators and Their Characterizations

Dual truncated Toeplitz operators and other restrictions of the multiplication by the independent variable $$M_z$$ M z on the classical $$L^2$$ L 2 space on the unit circle are investigated. Commutators are calculated and commutativity is characterized. A necessary and sufficient condition for any operator to be a dual truncated Toeplitz operator is established. A formula for recovering its symbol is stated.

Boundary behavior of Berezin symbols and related results

For a given function ? ? H? with |?(z)| < 1 (z ? D), we associate some special operators subspace and study some properties of these operators including behavior of their Berezin symbols. It turns that such boundary behavior is closely related to the Blaschke condition of sequences in the unit disk D of the complex plane. In terms of Berezin symbols the trace of some nuclear truncated Toeplitz operator is also calculated.

Reducing Subspaces of the Dual Truncated Toeplitz Operator

We define the dual truncated Toeplitz operators and give some basic properties of them. In particular, spectrum and reducing subspaces of some special dual truncated Toeplitz operator are characterized.