Reducing subspaces for strictly lower triangular operators

Abstract

In this paper, we establish a lattice isomorphism between the lattice of all reducing subspaces of a strictly lower triangular operator S and a sublattice of certain closed subspaces of $\ker S^{⁎}$. Here a strictly lower triangular operator means a bounded operator S on a Hilbert space H with ${\cap }_{n=0}^{\infty }\overline{S^n\left(H\right)}=0$, or equivalently, S possess a strictly lower triangular block matrix representation. Every unilateral operator-weighted shift is a strictly lower triangular operator. And many Toeplitz operators on function spaces are translations of strictly lower triangular operators. Further, we prove that every nonzero reducing subspace X of S satisfying $\dim X\cap \ker S^{⁎}<\infty$ can be expressed as a direct sum of at most $\dim X\cap \ker S^{⁎}$ minimal reducing subspaces. As an application, we characterize the reducing spaces of Toeplitz operators induced by quasi-homogeneous functions on n-analytic Bergman space. In particular, we show that $T_{z^q}$ with $q\ge 1$ on 2-analytic Bergman space has q minimal reducing subspaces. Moreover, we show that the von Neumann algebra generated by the commutants of $T_{z^q}$ and $T_{z^q}^{⁎}$, is ⁎-isomorphic to ${\oplus }_{k=1}^{q}C$.