Reducing subspaces of tensor products of weighted shifts

Abstract

A unilateral weighted shift A is said to be simple if its weight sequence {αn} satisfies ∇ 3 (α 2 n ) = 0 for all n 2. We prove that if A and B are two simple unilateral weighted shifts, then A ⊗ I + I ⊗ B is reducible if and only if A and B are unitarily equivalent. We also study the reducing subspaces of A k ⊗ I + I ⊗ B l and give some examples. As an application, we study the reducing subspaces of multiplication operators M z k +αw l on function spaces.

The graded structure induced by operators on a Hilbert space

“…Then H 2 (ω, δ) is a graded S-module, and Theorem 4.5 reveals that [1] S is a minimal reducing subspace. In this section, we mainly give a concise summary of the results in [12], [25]. There are two primary questions:…”

“…In [25], we proved that there is a class F of unilateral weighted shifts such that if A and B are both in F, then A ⊗ I + I ⊗ B is irreducible if and only if A and B are not unitarily equivalent. Can F be the whole class of unilateral weighted shifts, even a larger class?…”

“…Intuitively, the graded structures of Hilbert spaces that Hilbert polynomials can be defined on should attract some attentions. Lately, motivated by [12], [25], [32], we find that graded structure can be very useful in operator theory.…”

Multiplication operators on the Bergman space of bounded domains in C^d

“…This is equivalent to that {M φj,Ω : 1 ≤ j ≤ n} has no nonzero joint reducing subspace other than the whole space. In single-variable case, investigations on commutants and reducing subspaces of multiplication operators and von Neumann algebras induced by those operators have been done in [12,13,14,16,19,21,22,23,24,25,35,36,37] and some special multi-variable cases have been studied in [18,26,38,39].…”