Operator Theory in the Hardy Space over the Bidisk, III

Abstract

This paper is a continuation of an effort to build an organized operator theory in H 2 (D 2 ). It studies self-commutators for certain operator pairs and defines some numerical invariants for submodules. The fringe operator, which captures much of the information of the pairs, is defined in the last section and is used to establish an equality which connects the numerical invariants to traces of the self-commutators.

Two Inner Sequences Based Invariant Subspaces in $${{H}^{2} (\mathbb{D}^{2})}$$ H 2 ( D 2 )

“…It is indicated in [14] that if D is not contained in σ(S z ) or σ(S w ), both Σ 0 and Σ 1 are finite, moreover, Σ 0 = Σ 1 + 1 (cf. [15]).…”

Multiplication operators on the Bergman space via the Hardy space of the bidisk

“…Our main idea is to lift the Bergman shift up as the compression of a commuting pair of isometries on a nice subspace of the Hardy space of bidisk. This idea was used in studying the Hilbert modules by R. Douglas and V. Paulsen [12], operator theory in the Hardy space over the bidisk by R. Dougals and R. Yang [13], [37], [38] and [39]; the higher-order Hankel forms by S. Ferguson and R. Rochberg [10] and [11] and the lattice of the invariant subspaces of the Bergman shift by S. Richter [22].…”

Zero Based Invariant Subspaces and Fringe Operators Over the Bidisk

“…We denote by T z , T w the multiplication operators on H 2 by z, w. A nonzero closed subspace M of H 2 is said to be invariant if T z M ⊂ M and T w M ⊂ M . The structure of invariant subspaces of H 2 is fairly complicated and at this moment it seems to be out of reach (see [1,3,6,7]). We have…”

“…In [7], Yang studied the operator F M z on M ⊖ wM defined by F M z f = P M⊖wM T z f, f ∈ M ⊖ wM, where P A is the orthogonal projection from H 2 onto A ⊂ H 2 , and he called F M z the fringe operator of M .…”