Resultant operators for analytic functions in a simple-connected domain
Let \(\Gamma\) be a simple closed smooth contour in the complex plane \({\mathbb{C}}\) surrounding a bounded domain \(D^+\). For the family of analytic functions \(a_ 1(t),...,a_ n(t)\) in the domain \(D^+\) the authors introduce and study the singular integral operator \(R_{\alpha}=2(P_+-T_{\alpha}P_-)\), where \(P_{\pm}\) are Riesz projectors, \(T_{\alpha}\) is an analytic matrix-function \((b_{ij})_{i,j=1,...,n}\) with only (2n-1) non-zero elements: \(b_{in}=a_ i(t)\) for \(i=1,...,n\), and \(b_{ii}=t^{-\alpha_ i}\) for \(i=1,...,n=1\) \((\alpha_ i\in {\mathbb{R}},\alpha_ i>0).\) The main result: let \(N_ i(i=1,...,n-1)\) be the number of common zeroes of \(a_ n(t),a_ 1(t),...,a_ i(t)\) in \(D^+\), \(N_ 0\) is the number of zeroes of \(a_ n(t)\) in \(D^+\), \(\alpha_ n=N_ n=0\) and \(\alpha_ j-\alpha_{j+1}\geq N_ j-2N_ j+N_{j+1}\) \((j=1,2,...,n- 1)\); then ker \(R_{\alpha}\) has the following basis: \(\phi_ k(t)=[a_ 1(t),...,a_{n-1}(t),1+a_ n(t)]\cdot t^ k\cdot (d(t))^{- 1}(k=0,1,N_{n-1}-1)\), where d(t) is the polynomial greatest common divisor of \(a_ 1(z),a_ 2(z),...,a_ n(z)\) in \(D^+\); in particular, dim ker \(R_{\alpha}=N_{n-1}.\) The operator \(R_{\alpha}\) serves as a generalization of the classical Sylvester resultant matrix. This result is obtained as a corollary of an investigation of the Wiener-Hopf factorization problem for some classes of analytic matrix-functions.
Wiener-Hopf factorization problem for some classes of analytic matrix-functions
scientific article; zbMATH DE number 3979831