Resultant operators for analytic functions in a simple-connected domain
type: Item
instance of: scholarly article
title: Resultant operators for analytic functions in a simple-connected domain
zbMATH Open document ID: 0606.47030
DOI: 10.1007/BF01257064
author: B. A. Kon
author: Leonid Lerer
published in: Integral Equations and Operator Theory
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 45P05
Mathematics Subject Classification ID: 47Gxx
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 3979831
zbMATH Keywords: resultant operator
zbMATH Keywords: Bezout equation
zbMATH Keywords: singular integral operator
zbMATH Keywords: Sylvester resultant matrix
zbMATH Keywords: Wiener-Hopf factorization problem for some classes of analytic matrix-functions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3979831