Pub Date : 2023-11-21DOI: 10.1007/s00020-023-02752-5
Tianqiu Yu, Di Zhao, Li Qiu
In this paper, we first define the phases of a sectorial operator based on the numerical range. We are interested in the estimation of the spectrum phases of AB based on the phases of two sectorial operators A and B. Motivated by the classical small gain theorem, we formulate an operator small phase theorem with necessity for the invertibility of (I+AB), which plays a crucial role in feedback stability analysis. Afterwards, we consider the special class of sectorial operators of the form (P+K), where P is strictly positive and K is compact. More properties of the phases for those operators are studied, including those of compressions, Schur complements, operator means and products. Finally, for the special class of sectorial operators, we further establish a majorization relation between the phases of the spectrum of AB and the phases of two operators A and B.
{"title":"Phases of Sectorial Operators","authors":"Tianqiu Yu, Di Zhao, Li Qiu","doi":"10.1007/s00020-023-02752-5","DOIUrl":"https://doi.org/10.1007/s00020-023-02752-5","url":null,"abstract":"<p>In this paper, we first define the phases of a sectorial operator based on the numerical range. We are interested in the estimation of the spectrum phases of <i>AB</i> based on the phases of two sectorial operators <i>A</i> and <i>B</i>. Motivated by the classical small gain theorem, we formulate an operator small phase theorem with necessity for the invertibility of <span>(I+AB)</span>, which plays a crucial role in feedback stability analysis. Afterwards, we consider the special class of sectorial operators of the form <span>(P+K)</span>, where <i>P</i> is strictly positive and <i>K</i> is compact. More properties of the phases for those operators are studied, including those of compressions, Schur complements, operator means and products. Finally, for the special class of sectorial operators, we further establish a majorization relation between the phases of the spectrum of <i>AB</i> and the phases of two operators <i>A</i> and <i>B</i>.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138518210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-14DOI: 10.1007/s00020-023-02751-6
Michael T. Jury, Georgios Tsikalas
{"title":"Denjoy–Wolff Points on the Bidisk via Models","authors":"Michael T. Jury, Georgios Tsikalas","doi":"10.1007/s00020-023-02751-6","DOIUrl":"https://doi.org/10.1007/s00020-023-02751-6","url":null,"abstract":"","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"50 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134902650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-12DOI: 10.1007/s00020-023-02750-7
Martin Costabel, Monique Dauge, Khadijeh Nedaiasl
{"title":"Stability Analysis of a Simple Discretization Method for a Class of Strongly Singular Integral Equations","authors":"Martin Costabel, Monique Dauge, Khadijeh Nedaiasl","doi":"10.1007/s00020-023-02750-7","DOIUrl":"https://doi.org/10.1007/s00020-023-02750-7","url":null,"abstract":"","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"49 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135037590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-12DOI: 10.1007/s00020-023-02748-1
Masashi Wakaiki
Abstract This paper is concerned with the decay rate of $$e^{A^{-1}t}A^{-1}$$ eA-1tA-1 for the generator A of an exponentially stable $$C_0$$ C0 -semigroup on a Hilbert space. To estimate the decay rate of $$e^{A^{-1}t}A^{-1}$$ eA-1tA-1 , we apply a bounded functional calculus. Using this estimate and Lyapunov equations, we also study the quantified asymptotic behavior of the Crank-Nicolson scheme with smooth initial data. A similar argument is applied to a polynomially stable $$C_0$$ C0 -semigroup whose generator is normal.
研究Hilbert空间上指数稳定的$$C_0$$ c0 -半群的生成子A的衰减率($$e^{A^{-1}t}A^{-1}$$ e A - 1)和(A - 1)。为了估计$$e^{A^{-1}t}A^{-1}$$ e A - 1 t A - 1的衰减率，我们应用了有界泛函演算。利用这个估计和Lyapunov方程，我们还研究了具有光滑初始数据的Crank-Nicolson格式的量化渐近行为。一个类似的论证被应用于多项式稳定的$$C_0$$ C 0 -半群，它的生成器是正常的。
{"title":"Decay Rate of $$varvec{exp (A^{-1}t)A^{-1}}$$ on a Hilbert Space and the Crank–Nicolson Scheme with Smooth Initial Data","authors":"Masashi Wakaiki","doi":"10.1007/s00020-023-02748-1","DOIUrl":"https://doi.org/10.1007/s00020-023-02748-1","url":null,"abstract":"Abstract This paper is concerned with the decay rate of $$e^{A^{-1}t}A^{-1}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> for the generator A of an exponentially stable $$C_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -semigroup on a Hilbert space. To estimate the decay rate of $$e^{A^{-1}t}A^{-1}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , we apply a bounded functional calculus. Using this estimate and Lyapunov equations, we also study the quantified asymptotic behavior of the Crank-Nicolson scheme with smooth initial data. A similar argument is applied to a polynomially stable $$C_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -semigroup whose generator is normal.","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"26 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135037286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-07DOI: 10.1007/s00020-023-02749-0
Sam Farrington, Katie Gittins
Abstract We investigate the heat flow in an open, bounded set D in $$mathbb {R}^2$$ R2 with polygonal boundary $$partial D$$ ∂D . We suppose that D contains an open, bounded set $$widetilde{D}$$ D~ with polygonal boundary $$partial widetilde{D}$$ ∂D~ . The initial condition is the indicator function of $$widetilde{D}$$ D~ and we impose a Neumann boundary condition on the edges of $$partial D$$ ∂D . We obtain an asymptotic formula for the heat content of $$widetilde{D}$$ D~ in D as time $$tdownarrow 0$$ t↓0 .
{"title":"Heat Flow in Polygons with Reflecting Edges","authors":"Sam Farrington, Katie Gittins","doi":"10.1007/s00020-023-02749-0","DOIUrl":"https://doi.org/10.1007/s00020-023-02749-0","url":null,"abstract":"Abstract We investigate the heat flow in an open, bounded set D in $$mathbb {R}^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> with polygonal boundary $$partial D$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> . We suppose that D contains an open, bounded set $$widetilde{D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> with polygonal boundary $$partial widetilde{D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> </mml:math> . The initial condition is the indicator function of $$widetilde{D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> and we impose a Neumann boundary condition on the edges of $$partial D$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> . We obtain an asymptotic formula for the heat content of $$widetilde{D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> in D as time $$tdownarrow 0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>↓</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> .","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"316 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135474920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-07DOI: 10.1007/s00020-023-02747-2
Weisz Ferenc, Guangheng Xie, Dachun Yang
{"title":"Dyadic Maximal Operators on Martingale Musielak–Orlicz Hardy Type Spaces and Applications","authors":"Weisz Ferenc, Guangheng Xie, Dachun Yang","doi":"10.1007/s00020-023-02747-2","DOIUrl":"https://doi.org/10.1007/s00020-023-02747-2","url":null,"abstract":"","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"63 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135476598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-12DOI: 10.1007/s00020-023-02745-4
Youqing Ji, Yuanhang Zhang
{"title":"On the Power Set of Quasinilpotent Operators","authors":"Youqing Ji, Yuanhang Zhang","doi":"10.1007/s00020-023-02745-4","DOIUrl":"https://doi.org/10.1007/s00020-023-02745-4","url":null,"abstract":"","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135970084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-22DOI: 10.1007/s00020-023-02744-5
Lars Perlich
{"title":"Semiflows, Composition Semigroups, and the Approximation of Dirichlet-to-Neumann Semigroups","authors":"Lars Perlich","doi":"10.1007/s00020-023-02744-5","DOIUrl":"https://doi.org/10.1007/s00020-023-02744-5","url":null,"abstract":"","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136060442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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