Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk

Abstract

We consider the reducing subspaces of on 2 (D ), where ≥ 3, = 1 1 ⋅ ⋅ ⋅ , and ̸ = for ̸ = . We prove that each reducing subspace of is a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases that = 0 and ∈ (−1, +∞) \ Q, respectively. Finally, we give a complete description of minimal reducing subspaces of on 2 (D 3 ) with > −1.

Common reducing subspaces of several weighted shifts with operator weights

“…It turns out we can prove the converse of the above proposition if ρ is not a positive integer. The proof of the following lemma is more streamlined by comparing the roots of polynomials as Lemma 7 in [19], where reducing subspaces on weighted Bergman spaces on D 3 are discussed.…”

“…, N d ) and some of N i are distinct. If ω γ+kN /ω γ = ω δ+kN /ω δ for all k ≥ 0, the relationship between γ and δ could be complicated for d ≥ 3 as shown in [19] on K ρ (D d ). In particular, the reducing subspaces of T z N for N = (N 1 , N 2 , N 3 ) with distinct N i on K ρ (D d ) for d = 3, ρ > 1 are completely worked out there.…”

“…Recently, the reducing subspaces of some analytic Toeplitz operators on the Bergman space of the bidisk and polydisk were characterized in [15], [18], [14], and [19]. In [5], the author recovered the results from [21] and some results from [15] by studying the reducing subspaces of weighted shifts with operator weights.…”