Rongwei Yang

Operator theory in the Hardy space over the bidisk (II)

In this paper, we identify the vector valued Hardy space with the Hardy space over the bidisk and construct a universal model for thecontractive analytic functions. We will also study some elementary properties of the submodules and show, in some cases, how the operator theoretical properties are related to the module theoretical properties. The last part focus on the study of double commutativity of compression operators.

Inner sequence based invariant subspaces in

A closed subspace H 2 (D 2 ) is said to be invariant if it is invariant under the Toeplitz operators T z and T w . Invariant subspaces of H 2 (D 2 ) are well-known to be very complicated. So discovering some good examples of invariant subspaces will be beneficial to the general study. This paper studies a type of invariant subspace constructed through a sequence of inner functions. It will be shown that this type of invariant subspace has direct connections with the Jordan operator. Related calculations also give rise to a simple upper bound for Σ j 1 - |j|, where {λ j } are zeros of a Blaschke product.

PROJECTIVE SPECTRUM IN BANACH ALGEBRAS

For a tuple A = (A1, A2, …, An) of elements in a unital algebra over ℂ, its projective spectrumP(A) or p(A) is the collection of z ∈ ℂn, or respectively z ∈ ℙn-1 such that A(z) = z1A1 + z2A2 + ⋯ + znAn is not invertible in . In finite dimensional case, projective spectrum is a projective hypersurface. When A is commuting, P(A) looks like a bundle over the Taylor spectrum of A. In the case is reflexive or is a C*-algebra, the projective resolvent setPc(A) := ℂn \ P(A) is shown to be a disjoint union of domains of holomorphy. -valued 1-form A-1(z)dA(z) reveals the topology of Pc(A), and a Chern–Weil type homomorphism from invariant multilinear functionals to the de Rham cohomology is established.

BCL index and Fredholm tuples

In this paper we will prove that the BCL index for C*-algebras generated by two essentially doubly commuting isometries is equal to the index of the Fredholm tuples formed by these two isometries. We will then compute this index for certain sub-Hardy modules over the bidisk. Some interesting corollaries are also listed.

A trace formula for isometric pairs

It is well known that for every isometry V, tr[V*, V] = -ind(V). This fact for the shift operator is a basis for many important developments in operator theory and topology. In this paper we prove an analogous formula for a pair of isometries (V 1 , V 2 ), namely tr[V* 1 , V 1 , V* 2 , V 2 ) = -2ind(V 1 ,V 2 ), where [V* 1 , V 1 , V* 2 , V 2 ] is the complete anti-symmetric sum and ind(V 1 , V 2 ) is the Fredholm index of the pair (V 1 , V 2 ). The major tool is what we call the fringe operator. Two examples are considered.

Hermitian Geometry on Resolvent Set

For a tuple \({A} = ({A}_{1}, {A}_{2}, \ldots, {A}_{n})\) of elements in a unital Banach algebra \(\mathcal{B}\), its projective joint spectrum P(A) is the collection of \({z} \in {\mathbb{C}}^{n}\) such that \({A}(z) = {z}_{1}{A}_{1} + {z}_{2}{A}_{2} + \cdots + {z}_{n}{A}_{n}\) is not invertible. It is known that the \(\mathcal{B}\)-valued 1-form \({\omega}_{A}(z) = {A}^{-1}(z){dA}(z)\) contains much topological information about the joint resolvent set Pc(A). This paper studies geometric properties of Pc(A) with respect to Hermitian metrics defined through the \(\mathcal{B}\)-valued fundamental form \({\Omega}_{A} = -{\omega}^{\ast}_{A} \wedge {\omega}_{A}\) and its coupling with faithful states φ on \(\mathcal{B}\), i.e., φ(ΩA). The connection between the tuple A and the metric is the main subject of this paper. In particular, it shows that the Kahlerness of the metric is tied with the commutativity of the tuple, and its completeness is related to the Fuglede–Kadison determinant.

An index formula for the two variable Jordan block

On Hardy space over the bidisk, let be the compression of the pair to the quotient module . In this paper, we obtain an index formula for when it is Fredholm. It is also shown that the evaluation operator is compact on a Beurling type quotient module if and only if the corresponding inner function is a finite Blaschke product in .

A cross section from invariant subspaces to inner functions

Beurlings well known theorem connects the study of invariant subspaces to that of inner functions over the unit disc. In this paper, we will further explore this connection and, as a corollary of the result, show a one to one correspondence between the components of the invariant subspace lattice and the components of the space of inner functions.