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Reducing subspaces for the product of a forward and a backward operator-weighted shifts. (English) Zbl 07358770

Let \({\mathcal H}\) be a complex Hilbert space and let \({\mathbb Z}_+\) be the set of nonnegative integers. By \(\ell^2({\mathcal H})\) we denote the tensor product Hilbert space \({\mathcal H}\otimes \ell^2({\mathbb Z}_{+}^{2})\), where \(\ell^2({\mathbb Z}_{+}^{2})\) stands for the complex Hilbert space of complex sequences \(\bigl(\lambda_{(m,n)}\bigr)_{(m,n)\in {\mathbb Z}_{+}^{2}}\) such that \(\sum\limits_{(m,n)\in {\mathbb Z}_{+}^{2}}|\lambda_{(m,n)}|^2<\infty\). Let \(\bigl\{ e_{(m,n)};\; (m,n)\in {\mathbb Z}_{+}^{2}\bigr\}\) be the standard basis of \(\ell^2({\mathbb Z}_{+}^{2})\). Hilbert space \(\ell^2({\mathcal H})\) can be considered as the \({\mathcal H}\)-valued \(\ell^2({\mathbb Z}_{+}^{2})\) space; every vector \(h\in \ell^2({\mathcal H})\) is of the form \(h=\sum\limits_{(m,n)\in {\mathbb Z}_{+}^{2}} h_{(m,n)} e_{(m,n)}\), where \(\bigl(h_{(m,n)}\bigr)_{(m,n)\in {\mathbb Z}_{+}^{2}}\) is a sequence of vectors in \({\mathcal H}\) such that \(\sum\limits_{(m,n)\in {\mathbb Z}_{+}^{2}}\|h_{(m,n)}\|^2<\infty\).
Let \(\bigl(\Phi_{(m,n)}\bigr)_{(m,n)\in {\mathbb Z}_{+}^{2}}\) and \(\bigl(\Psi_{(m,n)}\bigr)_{(m,n)\in {\mathbb Z}_{+}^{2}}\) be two bounded sequences of bounded operators on \({\mathcal H}\) such that \(\Psi_{(m+1,n)}\Phi_{(m,n)}=\Phi_{(m,n+1)}\Psi_{(m,n)}\), for all \((m,n)\in {\mathbb Z}_{+}^{2}\). Then a pair of \(2\)-variable unilateral operator-weighted shifts consists of two bounded linear operators \(S_1\) and \(S_2\) on \(\ell^2({\mathcal H})\) which are defined by \[ S_1\biggl( \sum\limits_{(m,n)\in {\mathbb Z}_{+}^{2}} h_{(m,n)} e_{(m,n)}\biggr)= \sum\limits_{(m,n)\in {\mathbb Z}_{+}^{2}} (\Phi_{(m,n)}h_{(m,n)}) e_{(m+1,n)} \] and \[ S_2\biggl( \sum\limits_{(m,n)\in {\mathbb Z}_{+}^{2}} h_{(m,n)} e_{(m,n)}\biggr)= \sum\limits_{(m,n)\in {\mathbb Z}_{+}^{2}} (\Psi_{(m,n)}h_{(m,n)}) e_{(m,n+1)}. \]
The main results of the paper are related to the characterization of reducing subspaces of \(T\in \{S_1 S_{2}^{*}, S_{2}^{*} S_1\}\). For instance, it is proven that every reducing subspace \({\mathcal M}\) of \(T\) has the following wandering property: \[ {\mathcal M}=\mathrm{span}\bigl\{ T^k({\mathcal M}\ominus T{\mathcal M});\quad k\in {\mathbb Z}_+ \bigr\}. \] Reducing subspaces of Toeplitz operators induced by a non-analytic monomial on a weighted Hardy space of several variables are described, as well.

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A15 Invariant subspaces of linear operators
46E20 Hilbert spaces of continuous, differentiable or analytic functions
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