Hilbert spaces of \(q\)-shifted factorial series
For \(0<q<1\) and \(\alpha\geq 0\), Hilbert spaces \({\mathcal H}_ q^{(\alpha)}\) of functions analytic on \(\text{Re}(z)>(1-\alpha)/2\) are constructed with the inner product on \({\mathcal H}_ q^{(\alpha)}\) given by integration with respect to a finite measure \(\mu_ q^{(\alpha)}\) supported on \(\text{Re}(z)\geq (1-\alpha)/2\) and such that the functions \(e_ n(z;q)= (q^{1-z};q)_ n/ (q;q)_ n\) for \(n=0,1,2,\dots\) are an orthonormal set in \({\mathcal H}_ q^{(\alpha)}\). It is shown that the \(q\)-Cesàro matrix \(C_ q\) (the matrix whose \((i,j)\)th entry is \(q^ i(1-q)/ (1-q^{i+1})\) if \(i\geq j\) and 0 if \(i<j\) for \(i,j=0,1,2,\dots\)) can be represented as the restriction of a multiplication operator to the coinvariant subspace of \({\mathcal H}_ q^{(\alpha)}\) spanned by \(\{e_ n(z;q)\}_{n=0}^ \infty\). The properties of \({\mathcal H}_ q^{(\alpha)}\) (including inner product and reproducing kernel formulas) and of multiplication operators on \({\mathcal H}_ q^{(\alpha)}\) are studied and norm and spectral bounds are obtained for \(C_ q\).
scientific article; zbMATH DE number 483999