×

A factorization theorem for multiplier algebras of reproducing kernel Hilbert spaces. (English) Zbl 1278.46023

Let \((X,{\mathcal X},\mu)\) be a \(\sigma\)-finite measure space and let \(({\mathcal H},(.,.))\) be a Hilbert space with inner product \((.,.)\) and norm \(\|.\|\). If \(1\leq p<\infty\), then \(L^p(X)\) denotes the \(L^p\)-space \(\{[f]\) with \(f\) measurable on \(X:\|[f]\|_p=\|f\|_p<\infty\}\), where \([f]\) is the equivalence class of functions which differ from \(f\) on sets of measure 0, and \(\|f\|_p=(\int_X|f(x)|^p\,d\mu(x))^{1/p}\). If \({\mathcal H}\) is a space of complex-valued functions on \(X\), then the set of multipliers on \({\mathcal H}\), \({\mathcal M}({\mathcal H})\), is defined by \({\mathcal M}({\mathcal H})=\{ \)scalar-valued \(f\) on \({\mathcal H}:T_f(x)=fx\in{\mathcal H}\}\), with multiplier operator, \(T_f\),  \(T_f(x)=fx\). The set of bounded operators on \({\mathcal H}\) is denoted by \(B({\mathcal H})\). A weak*-closed subspace \(M\subseteq B({\mathcal H})\) is said to have property \((A_1(r))\) for some number \(r\geq 1\) if, for \(\varepsilon>0\) and \([L]\in Q_M\), the quotient space associated with the pre-annihilators of \(M\), there are vectors \(x,y\) in \({\mathcal H}\) such that \([L]=[x\otimes y]\), and \(\|x\|,\|y\|\leq((r+\varepsilon)\| L\|)^{1/2}\), and the subspace \(M\) is said to have property \((A_{\aleph_0}(r))\) for some \(r\geq 1\) if, for \(\varepsilon>0\) and each \(\{[L_{i,j}],~i,j=0,1,2,\dots\}\) in \(Q_M\) with \(K_i=\sum_{j\geq 0}\| [L_{i,j}]\|<\infty\), \(V_j=\sum_{i\geq 0}\|[L_{i,j}]\|<\infty\), there are vectors \(\{x_i,y_j,~i,j=0,1,2,\dots\}\) such that \([L_{i,j}]=[x_i\otimes y_j]\) and \(\|x_i\|\leq((r+\varepsilon)K_i)^{1/2}\), \(\|y_j\|\leq((r+ \varepsilon)V_j)^{1/2}\). The statements of the main theorem indicate that the dual algebra of multiplication operators on \({\mathcal H}\) has property \((A_1(1))\), and if the dual algebra does not have one-dimensional subspaces, then it has property \((A_{\aleph_0}(1))\).

MSC:

46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47L45 Dual algebras; weakly closed singly generated operator algebras
PDFBibTeX XMLCite
Full Text: DOI