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Commutative sequences of integrable functions and best approximation with respect to the weighted vector measure distance. (English) Zbl 1101.46028

In this paper, \((H, \langle\cdot,\cdot\rangle)\) is a real, separable Hilbert space, \(( \Omega, \Sigma)\) a measurable space, and \(\lambda : \Sigma\to H\) a countably additive measure with \(\| \lambda \|\) its semi-variation; \(\mathcal{C}\mathcal{A} (\Sigma, H) \) will denote the space of all such measures endowed with semi-variation norm. For \(f\in L_1(\lambda)\), \(f\lambda : \Sigma \to H\), \((f\lambda)(A)= \int_A f \,d\lambda\) is also in \(\mathcal{C} \mathcal{A} (\Sigma, H) \).
\(f\in L_2(\lambda)\) is called \(\lambda\)-positive if \( \lambda_f(A)=\langle\lambda(A), f^2\, d\lambda\rangle \) is a positive measure and controls \(\lambda\). For \(\lambda\)-positive \(f\), define \(L(\lambda, f)= \{g\in L_2(\lambda_f): fg \in L_{1}(\lambda) \}\) with norm \( \| g \|_{L( \lambda, f)} = \max \{ \| g \|_ {L_{2}(\lambda_{f})}, \| g \|_{f\lambda} \}\); \(L(\lambda, f) \subset L_1(f \lambda)\) and \(T_{f \lambda}\) denotes the restriction of the integration operator \(I_{f \lambda}\) to \(L(\lambda, f)\).
For a pair \( g, h\in L(\lambda, f)\), \(g(\lambda, f)\) commutes with \(h\) if \(\langle\int gf \,d\lambda, \int hf \,d\lambda\rangle = \int gh \,d \lambda_f\); its commutator \(c(g, h)\) is defined as \(c(g, h)= \int gh\, d \lambda_f-\langle \int gf\, d \lambda, \int hf\, d\lambda\rangle\).
\(\mathcal{B}\) denotes a pairwise \((\lambda, f)\)-commutative family of functions. A sequence \( \{ h_{i} \} \subset\text{span} \{ \mathcal{B} \} \) is said to be biorthonormal for \( \mathcal{B}\) if it is orthonormal with respect to \( \lambda_{f}\) and \( \overline{ T_{f \lambda} (\text{span} \{ h_{i}: i \in \mathbb N \})}= \overline{ T_{f \lambda} (\text{span} \{ \mathcal{B} \})}\). A biorthonormal sequence \( \{ h_{i} \} \subset \text{span} \{ \mathcal{B} \} \) is said to be complete if the operator \(P: L( \lambda, f) \to L( \lambda, f), \; P(g)= \sum_{i=1}^{\infty} (\int g h_{i}\, d \lambda_{f})h_{i} \) is continuous (in this case, it is a projection).
The distance function \( d_{\lambda, f}\) is defined in \(L( \lambda, f)\) by \(d_{\lambda, f} = \| \int (g-h)f\, d \lambda \|\).
The following are some of the major results:
A biorthonormal sequence \(\{ h_{i} \} \subset \text{span} \{ \mathcal{B} \}\) is complete if and only if the operator \(T: L( \lambda, f) \to \mathcal{C} \mathcal{A} (\Sigma, H) \), \( T(g)(A)= \sum_{i=1}^{\infty} (\int g h_{i}\, d \lambda_{f}) \int_{A} h_{i} f\, d \lambda\) is continuous.
Suppose that \( g \in L( \lambda, f)\) and that \(\{ h_{i} \} \subset \text{span} \{ \mathcal{B}\) } is a biorthonormal sequence. Then \( \alpha_{i} = \langle \int gf\, d \lambda, \int h_{i} f \,d \lambda \rangle\) is in \(\ell_{2}\) and the sequence of functions \( g_{n}= \sum_{i=1}^{n} \alpha_{i} h_{i}\) satisfies the conditions:
(1). For every \(n \in \mathbb N\), \( g_{n}\) is the unique function in span\(\{ h_{i}: 1 \leq i \leq n \}\) that attains the minimum \( \inf \{ \| \int (g-h)f d \lambda \|: h \in \text{span} \{ h_{i}: 1 \leq i \leq n \} \}\);
(2). If \(\{ g_{n} \}\) converges in \( L( \lambda, f)\) to \( h_{0}= \sum_{i=1}^{\infty} \alpha_{i} h_{i}\), then \( h_{0}\) is the unique function in \( \overline{ \text{span} \{ \mathcal{B} \}}\) that attains the minimum \( \inf \{ \| \int (g-h)f\, d \lambda \|: h \in \overline{ \text{span} \{ \mathcal{B} \}} \}\).
Let \(P\) be the associated projection for a complete biorthonormal system \( \{ h_{i} \}\) for \( \mathcal{B}\), \(g \in L( \lambda, f)\) and \( h \in \overline{ \text{span} \{ \mathcal{B} \}}\). Then \( c(g-h, g-h)= c(g, g) -2c(g- P(g), h)\).
Also, several other results are proved which establish some properties of \( L( \lambda, f)\), the commutative family of functions in \( L( \lambda, f)\), the commutator \(c\), and some relations between them.

MSC:

46G10 Vector-valued measures and integration
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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