Invertible factorization over multiplier algebras
A consequence of the classical Fejer-Riesz theorem says that, if \(T\) is a positive, invertible Toeplitz operator on the Hardy space \(H^2(\mathbb{D})\), then \(T = M^{*}_{a}M_{a}\), where \(a\) and \(\frac{1}{a}\) are bounded analytic functions on the unit disk. A vector-valued version of this result was established by \textit{D. B. Lowdenslager} [Ann. Math. (2) 78, 450--454 (1963; Zbl 0117.14701)]. In the paper under review, the author is concerned with such factorizations for more general positive, invertible operators acting on certain nice vector-valued reproducing kernel Hilbert spaces. A Hilbert space \(\mathcal{H}\) of analytic functions on a domain \(\Omega\subset\mathbb{C}^n\) is a reproducing kernel Hilbert space if point evaluation is bounded at every point in \(\Omega\). Such a space possesses a kernel function \(k\) and one usually denotes \(\mathcal{H}\) by \(H^2(k)\). If \(E\) is another Hilbert space, then \(H^2_{E}(k)\) is the tensor product \(H^2(k)\otimes E\), which consists of \(E\)-valued analytic functions. The author first proves, under the assumption that \(H^2(k)\) is nice (see the paper), a general abstract result which characterizes when a positive, invertible, bounded operator \(T\) on \(H^2_{E}(k)\) can be factored as \(T = A^{*}A\) with \(A\) and \(A^{-1}\) in \(\mathcal{M}(H^2_{E}(k))\), the multiplier algebra of \(H^2_{E}(k)\). The author then applies his theorem to derive interesting results on classical spaces such as the Bergman space over the unit disk and the Hardy space over the unit disk, polydisk, and the unit ball in higher dimensions. In each of these cases, a list of more concrete conditions is given. In the case of the Hardy space over the unit disk, the author recovers Lowdenslager's theorem which, as he states in the paper, provided the motivation for the main result.
vector-valued reproducing kernel Hilbert spaces
scientific article; zbMATH DE number 6153931