Fredholm property of partial differential operators of irregular singular type
type: Item
instance of: scholarly article
title: Fredholm property of partial differential operators of irregular singular type
zbMATH Open document ID: 0872.35025
DOI: 10.1007/BF02559712
author: Masatake Miyake
author: Masafumi Yoshino
published in: Arkiv för Matematik
publication date: 1997-10-15T00:00:00Z
review text: For \(w= (w_1,w_2)\), \(w_j>0\), \(s>0\) let \(G^s_w\) be the Gevrey space of formal series \(u(x)=\sum_{k\geq 0}u_k{x^k\over k!}\) such that \(\sum_{k\geq 0}u_k{x^k\over(k!)^s}\) is holomorphic in the bidisk \(\{(|x_1|<w_1)\), \((|x_2|< w_2)\}\). Consider the differential operator \[ P(x,D)= \sum_{\beta\in\mathbb{Z}^2,\gamma\in\mathbb{N}^2} a_{\gamma\beta}D^\beta x^\gamma, \] a finite sum with respect to \(\beta\). Define the Toeplitz symbol associated with \(P(x,D)\) by \[ L_{s,w}(z,\xi)= \sum_{|\beta|+ (1-s)(|\alpha|- |\beta|)= m}a_{\alpha\beta}z^{\alpha-\beta} w^{\alpha-\beta}\xi^\alpha,\quad \xi\in\mathbb{R}^2, \] where the \(s\)-Gevrey order, \(m=\sup_{\gamma,\beta}[|\beta|+ (1-s)(|\gamma|- |\beta|),\;a_{\gamma\beta}\neq 0]\) of \(P(x,D)\) is supposed to be finite. The main result of the paper is the following: The operator \(P:G^s_w\to G^s_w\) is a Fredholm operator of index zero if the following condition is satisfied \[ L_{s,w}(z,\xi)\neq 0\quad\text{for} \quad \xi\geq 0,\;|\xi|=1\quad\text{and} \quad |z_1|= |z_2|=1. \] This result is strongly related to a similar result found in the 70s by Kashiwara-Kawai-Sjöstrand.
reviewed by: Jan Janas
Mathematics Subject Classification ID: 35G05
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 894904
zbMATH Keywords: Gevrey space
zbMATH Keywords: Toeplitz symbol
MaRDI profile type: Publication
cites work: The Spectral Theory of Toeplitz Operators. (AM-99)
cites work: On a class of linear partial differential equations whose formal solutions always converge
cites work: Wiener-Hopf equation and Fredholm property of the Goursat problem in Gevrey space
cites work: Q4847096
cites work: Théorèmes d’indices Gevrey pour les équations différentielles ordinaires
full work available at URL: https://doi.org/10.1007/bf02559712
OpenAlex ID: W2079639690
description: scientific article; zbMATH DE number 894904