Integral Equations and Operator Theory最新文献

{"title":"Models of Holomorphic Functions on the Symmetrized Skew Bidisc.","authors":"Connor Evans, Zinaida A Lykova, N J Young","doi":"10.1007/s00020-026-02835-z","DOIUrl":"10.1007/s00020-026-02835-z","url":null,"abstract":"<p><p>The purpose of this paper is to develop the theory of holomorphic functions with modulus bounded by 1 on the symmetrized skew bidisc <dispformula> <math> <mrow><msub><mi>G</mi> <mi>r</mi></msub> <mover><mo>=</mo> <mtext>def</mtext></mover> <mrow><mo>{</mo></mrow> <mrow><mo>(</mo> <msub><mi>λ</mi> <mn>1</mn></msub> <mo>+</mo> <mi>r</mi> <msub><mi>λ</mi> <mn>2</mn></msub> <mo>,</mo> <mi>r</mi> <msub><mi>λ</mi> <mn>1</mn></msub> <msub><mi>λ</mi> <mn>2</mn></msub> <mo>)</mo></mrow> <mo>:</mo> <msub><mi>λ</mi> <mn>1</mn></msub> <mo>∈</mo> <mi>D</mi> <mo>,</mo> <msub><mi>λ</mi> <mn>2</mn></msub> <mo>∈</mo> <mi>D</mi> <mrow><mo>}</mo></mrow> <mo>,</mo></mrow> </math> </dispformula> for a fixed <math><mrow><mi>r</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo></mrow> </math> . We show the existence of a realization formula and a model formula for such holomorphic functions.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"98 2","pages":"18"},"PeriodicalIF":0.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13091886/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147728924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quadratic Subproduct Systems, Free Products, and Their C*-Algebras.","authors":"Francesca Arici, Yufan Ge","doi":"10.1007/s00020-025-02821-x","DOIUrl":"https://doi.org/10.1007/s00020-025-02821-x","url":null,"abstract":"<p><p>Motivated by the interplay between quadratic algebras, noncommutative geometry, and operator theory, we introduce the notion of quadratic subproduct systems of Hilbert spaces. Specifically, we study the subproduct systems induced by a finite number of complex quadratic polynomials in noncommuting variables, and describe their Toeplitz and Cuntz-Pimsner algebras. Inspired by the theory of graded associative algebras, we define a free product operation in the category of subproduct systems and show that this corresponds to the reduced free product of the Toeplitz algebras. Finally, we obtain results about the K-theory of the Toeplitz and Cuntz-Pimsner algebras of a large class of quadratic subproduct systems.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"98 2","pages":"19"},"PeriodicalIF":0.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13128699/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147814436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Double-Layer Potential for Spectral Constants Revisited.","authors":"Felix L Schwenninger, Jens de Vries","doi":"10.1007/s00020-025-02800-2","DOIUrl":"10.1007/s00020-025-02800-2","url":null,"abstract":"<p><p>We thoroughly analyse the double-layer potential's role in approaches to spectral sets in the spirit of Delyon-Delyon, Crouzeix and Crouzeix-Palencia. While the potential is well-studied, we aim to clarify on several of its aspects in light of these references. In particular, we illustrate how the associated integral operators can be used to characterize the convexity of the domain and the inclusion of the numerical range in its closure. We furthermore give a direct proof of a result by Putinar-Sandberg-a generalization of Berger-Stampfli's mapping theorem-circumventing dilation theory. Finally, we show for matrices that any smooth domain whose closure contains the numerical range admits a spectral constant only depending on the extremal function and vector. This constant is consistent with the so far best known absolute bound <math><mrow><mn>1</mn> <mo>+</mo> <msqrt><mn>2</mn></msqrt> </mrow> </math> .</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"97 2","pages":"13"},"PeriodicalIF":0.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12078368/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144093531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Function theory on the annulus in the dp-norm.","authors":"Jim Agler, Zinaida A Lykova, N J Young","doi":"10.1007/s00020-025-02814-w","DOIUrl":"https://doi.org/10.1007/s00020-025-02814-w","url":null,"abstract":"&lt;p&gt;&lt;p&gt;In this paper we shall use realization theory, a favourite technique of Rien Kaashoek, to prove new results about a class of holomorphic functions on an annulus &lt;dispformula&gt; &lt;math&gt; &lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;R&lt;/mi&gt; &lt;mi&gt;δ&lt;/mi&gt;&lt;/msub&gt; &lt;mover&gt;&lt;mo&gt;=&lt;/mo&gt; &lt;mtext&gt;def&lt;/mtext&gt;&lt;/mover&gt; &lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt; &lt;mi&gt;z&lt;/mi&gt; &lt;mo&gt;∈&lt;/mo&gt; &lt;mi&gt;C&lt;/mi&gt; &lt;mo&gt;:&lt;/mo&gt; &lt;mi&gt;δ&lt;/mi&gt; &lt;mo&gt;&lt;&lt;/mo&gt; &lt;mo&gt;|&lt;/mo&gt; &lt;mi&gt;z&lt;/mi&gt; &lt;mo&gt;|&lt;/mo&gt; &lt;mo&gt;&lt;&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt; &lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/math&gt; &lt;/dispformula&gt; where &lt;math&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;&lt;&lt;/mo&gt; &lt;mi&gt;δ&lt;/mi&gt; &lt;mo&gt;&lt;&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt; &lt;/math&gt; . The class of functions in question arises in the early work of R. G. Douglas and V. I. Paulsen on the rational dilation of a Hilbert space operator &lt;i&gt;T&lt;/i&gt; to a normal operator with spectrum in &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;msub&gt;&lt;mi&gt;R&lt;/mi&gt; &lt;mi&gt;δ&lt;/mi&gt;&lt;/msub&gt; &lt;/mrow&gt; &lt;/math&gt; . Their work suggested the following norm &lt;math&gt; &lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt; &lt;mo&gt;·&lt;/mo&gt; &lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt; &lt;mtext&gt;dp&lt;/mtext&gt;&lt;/msub&gt; &lt;/math&gt; on the space &lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;Hol&lt;/mtext&gt; &lt;mo&gt;(&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;R&lt;/mi&gt; &lt;mi&gt;δ&lt;/mi&gt;&lt;/msub&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;/math&gt; of holomorphic functions on &lt;math&gt;&lt;msub&gt;&lt;mi&gt;R&lt;/mi&gt; &lt;mi&gt;δ&lt;/mi&gt;&lt;/msub&gt; &lt;/math&gt; , &lt;dispformula&gt; &lt;math&gt; &lt;mrow&gt; &lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt; &lt;mi&gt;φ&lt;/mi&gt; &lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt; &lt;mtext&gt;dp&lt;/mtext&gt;&lt;/msub&gt; &lt;mover&gt;&lt;mo&gt;=&lt;/mo&gt; &lt;mtext&gt;def&lt;/mtext&gt;&lt;/mover&gt; &lt;mrow&gt;&lt;mo&gt;sup&lt;/mo&gt; &lt;mo&gt;{&lt;/mo&gt; &lt;mo&gt;‖&lt;/mo&gt; &lt;mi&gt;φ&lt;/mi&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;T&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;‖&lt;/mo&gt; &lt;mo&gt;:&lt;/mo&gt; &lt;mo&gt;‖&lt;/mo&gt; &lt;mi&gt;T&lt;/mi&gt; &lt;mo&gt;‖&lt;/mo&gt; &lt;mo&gt;≤&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt; &lt;msup&gt;&lt;mi&gt;T&lt;/mi&gt; &lt;mrow&gt;&lt;mo&gt;-&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt; &lt;/msup&gt; &lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt; &lt;mo&gt;≤&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt; &lt;mo&gt;/&lt;/mo&gt; &lt;mi&gt;δ&lt;/mi&gt; &lt;mspace&gt;&lt;/mspace&gt; &lt;mtext&gt;and&lt;/mtext&gt; &lt;mspace&gt;&lt;/mspace&gt; &lt;mi&gt;σ&lt;/mi&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;T&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;⊆&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;R&lt;/mi&gt; &lt;mi&gt;δ&lt;/mi&gt;&lt;/msub&gt; &lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt; &lt;/math&gt; &lt;/dispformula&gt; By analogy with the classical Schur class of holomorphic functions &lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt; with supremum norm at most 1 on the disc &lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt; , it is natural to consider the &lt;i&gt;dp-Schur class&lt;/i&gt; &lt;math&gt;&lt;msub&gt;&lt;mi&gt;S&lt;/mi&gt; &lt;mtext&gt;dp&lt;/mtext&gt;&lt;/msub&gt; &lt;/math&gt; of holomorphic functions of dp-norm at most 1 on &lt;math&gt;&lt;msub&gt;&lt;mi&gt;R&lt;/mi&gt; &lt;mi&gt;δ&lt;/mi&gt;&lt;/msub&gt; &lt;/math&gt; . Our central result is a Pick interpolation theorem for functions in &lt;math&gt;&lt;msub&gt;&lt;mi&gt;S&lt;/mi&gt; &lt;mtext&gt;dp&lt;/mtext&gt;&lt;/msub&gt; &lt;/math&gt; that is analogous to Abrahamse's Interpolation Theorem for bounded holomorphic functions on a multiply-connected domain. For a tuple &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mo&gt;(&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;λ&lt;/mi&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;mo&gt;⋯&lt;/mo&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;λ&lt;/mi&gt; &lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;/math&gt; of distinct interpolation nodes in &lt;math&gt;&lt;msub&gt;&lt;mi&gt;R&lt;/mi&gt; &lt;mi&gt;δ&lt;/mi&gt;&lt;/msub&gt; &lt;/math&gt; , we introduce a special set &lt;math&gt; &lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;G&lt;/mi&gt; &lt;mtext&gt;dp&lt;/mtext&gt;&lt;/msub&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;λ&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;/mrow&gt; &lt;/math&gt; of positive definite &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;×&lt;/mo&gt; &lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt; &lt;/math&gt; matrices, which we call &lt;i&gt;DP Szegő kerne","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"97 4","pages":"30"},"PeriodicalIF":0.9,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12580444/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145443968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"$$varvec{q}$$ -rational Functions and Interpolation with Complete Nevanlinna–Pick Kernels","authors":"Daniel Alpay, Paula Cerejeiras, Uwe Kaehler, Baruch Schneider","doi":"10.1007/s00020-024-02779-2","DOIUrl":"https://doi.org/10.1007/s00020-024-02779-2","url":null,"abstract":"<p>In this paper we introduce the concept of matrix-valued <i>q</i>-rational functions. In comparison to the classical case, we give different characterizations with principal emphasis on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna–Pick kernels in the context of <i>q</i>-deformed reproducing kernel Hilbert spaces and provide first applications in terms of an interpolation problem using Schur multipliers and complete Nevanlinna–Pick kernels.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"151 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256783","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}

{"title":"Mosco Convergence of Gradient Forms with Non-Convex Interaction Potential","authors":"Martin Grothaus, Simon Wittmann","doi":"10.1007/s00020-024-02775-6","DOIUrl":"https://doi.org/10.1007/s00020-024-02775-6","url":null,"abstract":"<p>This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, <span>({mathcal {E}}^N)</span> on <span>(L^2(E,mu _N))</span> for <span>(Nin {mathbb {N}})</span>, in the framework of converging Hilbert spaces by K. Kuwae and T. Shioya. The basic assumption is weak measure convergence of the family <span>({(mu _N)}_{N})</span> on the state space <i>E</i>—either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on <span>({(mu _N)}_{N})</span> try to impose as little restrictions as possible. The problem has fully been solved if the family <span>({(mu _N)}_{N})</span> contain only log-concave measures, due to Ambrosio et al. (Probab Theory Relat. Fields 145:517–564, 2009). However, for a large class of convergence problems the assumption of log-concavity fails. The article suggests a way to overcome this hindrance, as it presents a new approach. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result by Bounebache and Zambotti (J Theor Probab 27:168–201, 2014).\u0000</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"17 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256437","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}

{"title":"Logarithmically Enhanced Area-Laws for Fermions in Vanishing Magnetic Fields in Dimension Two","authors":"Paul Pfeiffer, Wolfgang Spitzer","doi":"10.1007/s00020-024-02778-3","DOIUrl":"https://doi.org/10.1007/s00020-024-02778-3","url":null,"abstract":"<p>We consider fermionic ground states of the Landau Hamiltonian, <span>(H_B)</span>, in a constant magnetic field of strength <span>(B&gt;0)</span> in <span>({mathbb {R}}^2)</span> at some fixed Fermi energy <span>(mu &gt;0)</span>, described by the Fermi projection <span>(P_B:=1(H_Ble mu ))</span>. For some fixed bounded domain <span>(Lambda subset {mathbb {R}}^2)</span> with boundary set <span>(partial Lambda )</span> and an <span>(L&gt;0)</span> we restrict these ground states spatially to the scaled domain <span>(L Lambda )</span> and denote the corresponding localised Fermi projection by <span>(P_B(LLambda ))</span>. Then we study the scaling of the Hilbert-space trace, <span>(textrm{tr} f(P_B(LLambda )))</span>, for polynomials <i>f</i> with <span>(f(0)=f(1)=0)</span> of these localised ground states in the joint limit <span>(Lrightarrow infty )</span> and <span>(Brightarrow 0)</span>. We obtain to leading order logarithmically enhanced area-laws depending on the size of <i>LB</i>. Roughly speaking, if 1/<i>B</i> tends to infinity faster than <i>L</i>, then we obtain the known enhanced area-law (by the Widom–Sobolev formula) of the form <span>(L ln (L) a(f,mu ) |partial Lambda |)</span> as <span>(Lrightarrow infty )</span> for the (two-dimensional) Laplacian with Fermi projection <span>(1(H_0le mu ))</span>. On the other hand, if <i>L</i> tends to infinity faster than 1/<i>B</i>, then we get an area law with an <span>(L ln (mu /B) a(f,mu ) |partial Lambda |)</span> asymptotic expansion as <span>(Brightarrow 0)</span>. The numerical coefficient <span>(a(f,mu ))</span> in both cases is the same and depends solely on the function <i>f</i> and on <span>(mu )</span>. The asymptotic result in the latter case is based upon the recent joint work of Leschke, Sobolev and the second named author [7] for fixed <i>B</i>, a proof of the sine-kernel asymptotics on a global scale, and on the enhanced area-law in dimension one by Landau and Widom. In the special but important case of a quadratic function <i>f</i> we are able to cover the full range of parameters <i>B</i> and <i>L</i>. In general, we have a smaller region of parameters (<i>B</i>, <i>L</i>) where we can prove the two-scale asymptotic expansion <span>(textrm{tr} f(P_B(LLambda )))</span> as <span>(Lrightarrow infty )</span> and <span>(Brightarrow 0)</span>.\u0000</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"13 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256785","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}

{"title":"Positive Semidefinite Maps on $$*$$ -Semigroupoids and Linearisations","authors":"Aurelian Gheondea, Bogdan Udrea","doi":"10.1007/s00020-024-02777-4","DOIUrl":"https://doi.org/10.1007/s00020-024-02777-4","url":null,"abstract":"<p>Motivated by current investigations in dilation theory, in both operator theory and operator algebras, and the theory of groupoids, we obtain a generalisation of the Sz-Nagy’s Dilation Theorem for operator valued positive semidefinite maps on <span>(*)</span>-semigroupoids with unit, with varying degrees of aggregation, firstly by <span>(*)</span>-representations with unbounded operators and then we characterise the existence of the corresponding <span>(*)</span>-representations by bounded operators. By linearisation of these constructions, we obtain similar results for operator valued positive semidefinite maps on <span>(*)</span>-algebroids with unit and then, for the special case of <span>(B^*)</span>-algebroids with unit, we obtain a generalisation of the Stinespring’s Dilation Theorem. As an application of the generalisation of the Stinespring’s Dilation Theorem, we show that some natural questions on <span>(C^*)</span>-algebroids are equivalent.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"15 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217774","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}

{"title":"$$C^*$$ -Algebras Associated to Transfer Operators for Countable-to-One Maps","authors":"Krzysztof Bardadyn, Bartosz K. Kwaśniewski, Andrei V. Lebedev","doi":"10.1007/s00020-024-02774-7","DOIUrl":"https://doi.org/10.1007/s00020-024-02774-7","url":null,"abstract":"<p>Our initial data is a transfer operator <i>L</i> for a continuous, countable-to-one map <span>(varphi :Delta rightarrow X)</span> defined on an open subset of a locally compact Hausdorff space <i>X</i>. Then <i>L</i> may be identified with a ‘potential’, i.e. a map <span>(varrho :Delta rightarrow X)</span> that need not be continuous unless <span>(varphi )</span> is a local homeomorphism. We define the crossed product <span>(C_0(X)rtimes L)</span> as a universal <span>(C^*)</span>-algebra with explicit generators and relations, and give an explicit faithful representation of <span>(C_0(X)rtimes L)</span> under which it is generated by weighted composition operators. We explain its relationship with Exel–Royer’s crossed products, quiver <span>(C^*)</span>-algebras of Muhly and Tomforde, <span>(C^*)</span>-algebras associated to complex or self-similar dynamics by Kajiwara and Watatani, and groupoid <span>(C^*)</span>-algebras associated to Deaconu–Renault groupoids. We describe spectra of core subalgebras of <span>(C_0(X)rtimes L)</span>, prove uniqueness theorems for <span>(C_0(X)rtimes L)</span> and characterize simplicity of <span>(C_0(X)rtimes L)</span>. We give efficient criteria for <span>(C_0(X)rtimes L)</span> to be purely infinite simple and in particular a Kirchberg algebra.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"23 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217778","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}

{"title":"The Heisenberg Group Action on the Siegel Domain and the Structure of Bergman Spaces","authors":"Julio A. Barrera-Reyes, Raúl Quiroga-Barranco","doi":"10.1007/s00020-024-02776-5","DOIUrl":"https://doi.org/10.1007/s00020-024-02776-5","url":null,"abstract":"<p>We study the biholomorphic action of the Heisenberg group <span>(mathbb {H}_n)</span> on the Siegel domain <span>(D_{n+1})</span> (<span>(n ge 1)</span>). Such <span>(mathbb {H}_n)</span>-action allows us to obtain decompositions of both <span>(D_{n+1})</span> and the weighted Bergman spaces <span>(mathcal {A}^2_lambda (D_{n+1}))</span> (<span>(lambda &gt; -1)</span>). Through the use of symplectic geometry we construct a natural set of coordinates for <span>(D_{n+1})</span> adapted to <span>(mathbb {H}_n)</span>. This yields a useful decomposition of the domain <span>(D_{n+1})</span>. The latter is then used to compute a decomposition of the Bergman spaces <span>(mathcal {A}^2_lambda (D_{n+1}))</span> (<span>(lambda &gt; -1)</span>) as direct integrals of Fock spaces. This effectively shows the existence of an interplay between Bergman spaces and Fock spaces through the Heisenberg group <span>(mathbb {H}_n)</span>. As an application, we consider <span>(mathcal {T}^{(lambda )}(L^infty (D_{n+1})^{mathbb {H}_n}))</span> the <span>(C^*)</span>-algebra acting on the weighted Bergman space <span>(mathcal {A}^2_lambda (D_{n+1}))</span> (<span>(lambda &gt; -1)</span>) generated by Toeplitz operators whose symbols belong to <span>(L^infty (D_{n+1})^{mathbb {H}_n})</span> (essentially bounded and <span>(mathbb {H}_n)</span>-invariant). We prove that <span>(mathcal {T}^{(lambda )}(L^infty (D_{n+1})^{mathbb {H}_n}))</span> is commutative and isomorphic to <span>(textrm{VSO}(mathbb {R}_+))</span> (very slowly oscillating functions on <span>(mathbb {R}_+)</span>), for every <span>(lambda &gt; -1)</span> and <span>(n ge 1)</span>.</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"9 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217787","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}