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One-step extensions of subnormal 2-variable weighted shifts. (English) Zbl 1318.47033

The authors consider the following problem. Assume that \(W_{(\alpha,\beta)}|_\mathcal{M}\) and \(W_{(\alpha,\beta)}|_\mathcal{N}\) are subnormal with Berger measures \(\mu_\mathcal{M}\) and \(\mu_\mathcal{N}\), respectively. Find necessary and sufficient conditions on \(\mu_\mathcal{M}\), \(\mu_\mathcal{N}\) and \(\beta_{00}\) for the subnormality of \(W_{(\alpha,\beta)}\). Consider double-indexed positive bounded sequences \(\alpha \equiv \{ \alpha_{(k_1,k_2)}\}\), \(\beta \equiv \{ \beta_{(k_1,k_2)}\} \in \ell^\infty(\mathbb{Z}_+^2)\), \((k_1, k_2) \in \mathbb{Z}_+^2 := \mathbb{Z}_+ \times \mathbb{Z}_+\) and let \(\ell^2(\mathbb{Z}_+^2)\) be the Hilbert space of square-summable complex sequences indexed by \(\mathbb{Z}_+^2\). Let \(W_{(\alpha,\beta)} \equiv (T_1, T_2)\) be the 2-variable weighted shift. Let \(\mathcal{M}\) (resp., \(\mathcal{N}\)) be the invariant subspace of \(\ell^2(\mathbb{Z}_+^2)\) spanned by the canonical orthonormal basis associated to indices \({\mathbf k}=(k_1,k_2)\) with \(k_1 \geq 0\) and \(k_2 \geq 1\) (resp., \(k_1 \geq 1\) and \(k_2 \geq 0\)). First, assume that \(W_{(\alpha,\beta)}|_\mathcal{M}\) and \(W_{(\alpha,\beta)}|_\mathcal{N}\) are subnormal with Berger measures \(\mu_\mathcal{M}\) and \(\mu_\mathcal{N}\), respectively, and let \(c:=\frac{\int s d\mu_\mathcal{M}}{\int t d\mu_\mathcal{N}} \equiv \frac{\alpha_{01}^2}{\beta_{10}^2}\). Then the authors prove that \(W_{(\alpha,\beta)}\) is subnormal if and only if the following four conditions hold:
(i) \(\frac{1}{t} \in L^1(\mu_\mathcal{M})\);
(ii) \(\frac{1}{s} \in L^1(\mu_\mathcal{N})\);
(iii) \(c\beta_{00}^2\|\frac{1}{s}\|_{L^1(\mu_\mathcal{N})} \leq 1\);
(iv) \(\beta_{00}^2 \{ \|\frac{1}{t}\|_{L^1(\mu_\mathcal{M})}(\mu_\mathcal{M})_{ext}^X + c\|\frac{1}{s}\|_{L^1(\mu_\mathcal{N})}\delta_0 -\frac{c}{s}(\mu_\mathcal{N})^X \} \leq \delta_0\).
Next, assume that \(W_{(\alpha,\beta)}|_\mathcal{M}\) and \(W_{(\alpha,\beta)}|_\mathcal{N}\) are subnormal with Berger measures \(\mu_\mathcal{M}\) and \(\mu_\mathcal{N}\), respectively, and let \(\rho := \mu_{\mathcal{M}}^X\), i.e., \(\rho\) is the Berger measure of shift\((\alpha_{01}, \alpha_{11}, \dots{})\). Also, assume that \(\mu_{\mathcal{M} \cap \mathcal{N}} = \xi \times \eta\) for some 1-variable probability measures \(\xi\) and \(\eta\). Then the authors prove that \(W_{(\alpha,\beta)}\) is subnormal if and only if the following conditions hold:
(i) \(\frac{1}{t} \in L^1(\mu_\mathcal{M})\);
(ii) \(\beta_{00}^2\|\frac{1}{t}\|_{L^1(\mu_\mathcal{M})} \leq 1\);
(iii) \((\beta_{00}^2\|\frac{1}{t}\|_{L^1(\tau_1)})\rho = (\beta_{00}^2\|\frac{1}{t}\|_{L^1(\mu_\mathcal{M})})\rho \leq \sigma\), where \(\tau_1\) denotes the Berger measure of shift\((\beta_{01}, \beta_{02}, \dots{})\) and \(\sigma\) denotes the Berger measure of shift\((\alpha_{00}, \alpha_{10}, \dots{})\).

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
28A50 Integration and disintegration of measures
44A60 Moment problems
47A20 Dilations, extensions, compressions of linear operators
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