Pub Date : 2024-01-25DOI: 10.1007/s00020-023-02746-3
Branko Ćurgus, Volodymyr Derkach, Carsten Trunk
We consider the indefinite Sturm–Liouville differential expression
$$begin{aligned} {mathfrak {a}}(f):= - frac{1}{w}left( frac{1}{r} f' right) ', end{aligned}$$
where ({mathfrak {a}}) is defined on a finite or infinite open interval I with (0in I) and the coefficients r and w are locally summable and such that r(x) and (({text {sgn}},x) w(x)) are positive a.e. on I. With the differential expression ({mathfrak {a}}) we associate a nonnegative self-adjoint operator A in the Krein space (L^2_w(I)) which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of I with the positive and the negative semi-axis. For the operator A we derive conditions in terms of the coefficients w and r for the existence of a Riesz basis consisting of generalized eigenfunctions of A and for the similarity of A to a self-adjoint operator in a Hilbert space (L^2_{|w|}(I)). These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces which are couplings of two symmetric operators acting in Hilbert spaces.
我们考虑不确定的 Sturm-Liouville 微分表达式 $$begin{aligned} {mathfrak {a}}(f)：= - frac{1}{w}left( frac{1}{r} f' right) ', end{aligned}$$其中({mathfrak {a}})定义在有限或无限开区间I上，且(0in I) 和系数r和w是局部可求和的，并且使得r(x)和(({text {sgn}},x) w(x))是正的。通过微分表达式 ({mathfrak {a}})，我们在克雷因空间 (L^2_w(I)) 中关联了一个非负自相关算子 A，它被视为希尔伯特空间中对称算子的耦合，与 I 与正半轴和负半轴的交点相关。对于算子 A，我们从系数 w 和 r 的角度推导出存在由 A 的广义特征函数组成的里兹基的条件，以及 A 与希尔伯特空间 (L^2_{|w|}(I))中的自交算子相似的条件。这些结果是关于克雷因空间中非负自相关算子临界点正则性的抽象结果的后果，而克雷因空间是作用于希尔伯特空间的两个对称算子的耦合。
{"title":"Indefinite Sturm–Liouville Operators in Polar Form","authors":"Branko Ćurgus, Volodymyr Derkach, Carsten Trunk","doi":"10.1007/s00020-023-02746-3","DOIUrl":"https://doi.org/10.1007/s00020-023-02746-3","url":null,"abstract":"<p>We consider the indefinite Sturm–Liouville differential expression </p><span>$$begin{aligned} {mathfrak {a}}(f):= - frac{1}{w}left( frac{1}{r} f' right) ', end{aligned}$$</span><p>where <span>({mathfrak {a}})</span> is defined on a finite or infinite open interval <i>I</i> with <span>(0in I)</span> and the coefficients <i>r</i> and <i>w</i> are locally summable and such that <i>r</i>(<i>x</i>) and <span>(({text {sgn}},x) w(x))</span> are positive a.e. on <i>I</i>. With the differential expression <span>({mathfrak {a}})</span> we associate a nonnegative self-adjoint operator <i>A</i> in the Krein space <span>(L^2_w(I))</span> which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of <i>I</i> with the positive and the negative semi-axis. For the operator <i>A</i> we derive conditions in terms of the coefficients <i>w</i> and <i>r</i> for the existence of a Riesz basis consisting of generalized eigenfunctions of <i>A</i> and for the similarity of <i>A</i> to a self-adjoint operator in a Hilbert space <span>(L^2_{|w|}(I))</span>. These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces which are couplings of two symmetric operators acting in Hilbert spaces.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139552743","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 : 2023-12-10DOI: 10.1007/s00020-023-02742-7
Joelle Jreis, Pascal Lefèvre
We characterize the integration operators (V_g) with symbol g for which (V_g) acts as an absolutely summing operator on weighted Bloch spaces (mathcal {B}^{beta }) and on weighted Bergman spaces (mathscr {A}^p_alpha ). We show that (V_g) is r-summing on (mathscr {A}^p_alpha ), (1 le p <infty ), if and only if g belongs to a suitable Besov space. We also show that there is no non trivial nuclear Volterra operators (V_g) on Bloch spaces and on Bergman spaces.
我们描述了符号为 g 的积分算子 (V_g)，对于这些算子，(V_g) 在加权布洛赫空间 (mathcal {B}^{beta }) 和加权伯格曼空间 (mathscr {A}^p_alpha )上作为绝对求和算子。我们证明，当且仅当g属于一个合适的贝索夫空间时，(V_g)在(mathscr {A}^p_alpha )、(1 le p <infty )上是r求和的。我们还证明了在布洛赫空间和贝格曼空间上不存在非微不足道的核 Volterra 算子 (V_g)。
{"title":"Some Operator Ideal Properties of Volterra Operators on Bergman and Bloch Spaces","authors":"Joelle Jreis, Pascal Lefèvre","doi":"10.1007/s00020-023-02742-7","DOIUrl":"https://doi.org/10.1007/s00020-023-02742-7","url":null,"abstract":"<p>We characterize the integration operators <span>(V_g)</span> with symbol <i>g</i> for which <span>(V_g)</span> acts as an absolutely summing operator on weighted Bloch spaces <span>(mathcal {B}^{beta })</span> and on weighted Bergman spaces <span>(mathscr {A}^p_alpha )</span>. We show that <span>(V_g)</span> is <i>r</i>-summing on <span>(mathscr {A}^p_alpha )</span>, <span>(1 le p <infty )</span>, if and only if <i>g</i> belongs to a suitable Besov space. We also show that there is no non trivial nuclear Volterra operators <span>(V_g)</span> on Bloch spaces and on Bergman spaces.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"7 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138560805","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 : 2023-11-21DOI: 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}
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