Hilbert spaces of vector-valued functions generated by quadratic forms (Q1947295)

Let \(E\subset\mathbb{R}^d\) be a measurable subset with positive Lebesgue measure. Let \(G(t)\) be a Hermitian positive-definite matrix-valued function defined on the set \(E\). Let \(L^2(E;G)\) denote the Hilbert space of functions that are integrable on \(E\), taking values in \(\mathbb{C}^d\), such that \[ \left\| f\right\|_{L^2(E;G)}^2=\int_E \left\langle G(t) f(t), f(t)\right\rangle_{\mathbb{C}^d} dt<\infty. \] One of the main goals of this paper is to show that certain basis properties of collections of functions behave well in this weighted setting. The basis conditions of interest (e.g., Schauder, unconditionality, minimality) are expressed in terms of invertibility and \(A_2\)-type conditions on the diagonal entries of the matrix \(G\) that is playing the role of the weight. The interested reader should consult the paper ``Wavelets and the angle between past and future'' by \textit{S. Treil} and \textit{A. Volberg} [J. Funct. Anal. 143, No. 2, 269--308 (1997; Zbl 0876.42027)] where the theory of ``matrix \(A_2\)'' is developed and, in particular, the Haar basis is shown to be unconditional for \(L^2(\mathbb{R};W)\) if and only if the matrix \(W\) satisfies a matricial analogue of the \(A_2\) condition. This result was later extended to \(L^p\) by \textit{F. L. Nazarov} and \textit{S. P. Treil'} in [St. Petersbg. Math. J. 8, No. 5, 721--824 (1997); translation from Algebra Anal. 8, No. 5, 32--132 (1996; Zbl 0873.42011)]. The interested reader can also consult ``Matrix \(A_p\) weights via \(S\)-functions'' by \textit{A. Volberg} [J. Am. Math. Soc. 10, No. 2, 445--466 (1997; Zbl 0877.42003)]. It is very likely that the results obtained in the paper under review can be derived from the earlier results by Nazarov, Treil and Volberg.