{"title":"On the Spectral Properties of Selfadjoint Partial Integral Operators with a Nondegenerate Kernel","authors":"","doi":"10.1134/s0037446624020204","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We consider bounded selfadjoint linear integral operators <span> <span>\\( T_{1} \\)</span> </span> and <span> <span>\\( T_{2} \\)</span> </span> in the Hilbert space <span> <span>\\( L_{2}([a,b]\\times[c,d]) \\)</span> </span> which are usually called partial integral operators. We assume that <span> <span>\\( T_{1} \\)</span> </span> acts on a function <span> <span>\\( f(x,y) \\)</span> </span> in the first argument and performs integration in <span> <span>\\( x \\)</span> </span>, while <span> <span>\\( T_{2} \\)</span> </span> acts on <span> <span>\\( f(x,y) \\)</span> </span> in the second argument and performs integration in <span> <span>\\( y \\)</span> </span>. We assume further that <span> <span>\\( T_{1} \\)</span> </span> and <span> <span>\\( T_{2} \\)</span> </span> are bounded but not compact, whereas <span> <span>\\( T_{1}T_{2} \\)</span> </span> is compact and <span> <span>\\( T_{1}T_{2}=T_{2}T_{1} \\)</span> </span>. Partial integral operators arise in various areas of mechanics, the theory of integro-differential equations, and the theory of Schrödinger operators. We study the spectral properties of <span> <span>\\( T_{1} \\)</span> </span>, <span> <span>\\( T_{2} \\)</span> </span>, and <span> <span>\\( T_{1}+T_{2} \\)</span> </span> with nondegenerate kernels and established some formula for the essential spectra of <span> <span>\\( T_{1} \\)</span> </span> and <span> <span>\\( T_{2} \\)</span> </span>. Furthermore, we demonstrate that the discrete spectra of <span> <span>\\( T_{1} \\)</span> </span> and <span> <span>\\( T_{2} \\)</span> </span> are empty, and prove a theorem on the structure of the essential spectrum of <span> <span>\\( T_{1}+T_{2} \\)</span> </span>. Also, under study is the problem of existence of countably many eigenvalues in the discrete spectrum of <span> <span>\\( T_{1}+T_{2} \\)</span> </span>.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":"21 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Siberian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0037446624020204","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider bounded selfadjoint linear integral operators \( T_{1} \) and \( T_{2} \) in the Hilbert space \( L_{2}([a,b]\times[c,d]) \) which are usually called partial integral operators. We assume that \( T_{1} \) acts on a function \( f(x,y) \) in the first argument and performs integration in \( x \), while \( T_{2} \) acts on \( f(x,y) \) in the second argument and performs integration in \( y \). We assume further that \( T_{1} \) and \( T_{2} \) are bounded but not compact, whereas \( T_{1}T_{2} \) is compact and \( T_{1}T_{2}=T_{2}T_{1} \). Partial integral operators arise in various areas of mechanics, the theory of integro-differential equations, and the theory of Schrödinger operators. We study the spectral properties of \( T_{1} \), \( T_{2} \), and \( T_{1}+T_{2} \) with nondegenerate kernels and established some formula for the essential spectra of \( T_{1} \) and \( T_{2} \). Furthermore, we demonstrate that the discrete spectra of \( T_{1} \) and \( T_{2} \) are empty, and prove a theorem on the structure of the essential spectrum of \( T_{1}+T_{2} \). Also, under study is the problem of existence of countably many eigenvalues in the discrete spectrum of \( T_{1}+T_{2} \).
期刊介绍:
Siberian Mathematical Journal is journal published in collaboration with the Sobolev Institute of Mathematics in Novosibirsk. The journal publishes the results of studies in various branches of mathematics.