Pub Date : 2023-11-16DOI: 10.1007/s43037-023-00308-8
Kaikai Han, Yanyan Tang
It is known that the invariant subspace problem for Hilbert spaces is equivalent to the statement that all minimal non-trivial invariant subspaces for a universal operator are one dimensional. In this paper, we first give a characterization of the boundedness of composition operators on weighted Dirichlet spaces ({mathcal {D}}_{alpha }(Pi ^{+})) over the upper half-plane (Pi ^{+}) using generalized Nevanlinna counting functions, where (alpha >-1.) As an application, we discuss the boundedness of composition operators on ({mathcal {D}}_{alpha }(Pi ^{+})) induced by linear fractional self-maps of (Pi ^{+}.) Second, we characterize composition operators and their adjoints induced by affine self-maps of (Pi ^{+}) that have universal translates on ({mathcal {D}}_{alpha }(Pi ^{+}).) Moreover, we investigate which composition operators and their adjoints induced by hyperbolic non-automorphism self-maps of the open unit disk ({mathbb {D}}) have universal translates on weighted Dirichlet spaces ({mathcal {D}}_{alpha }({mathbb {D}})) for (alpha >-1.) Finally, we consider the minimal invariant subspaces of the composition operators that have universal translates.
{"title":"Universal composition operators on weighted Dirichlet spaces","authors":"Kaikai Han, Yanyan Tang","doi":"10.1007/s43037-023-00308-8","DOIUrl":"https://doi.org/10.1007/s43037-023-00308-8","url":null,"abstract":"<p>It is known that the invariant subspace problem for Hilbert spaces is equivalent to the statement that all minimal non-trivial invariant subspaces for a universal operator are one dimensional. In this paper, we first give a characterization of the boundedness of composition operators on weighted Dirichlet spaces <span>({mathcal {D}}_{alpha }(Pi ^{+}))</span> over the upper half-plane <span>(Pi ^{+})</span> using generalized Nevanlinna counting functions, where <span>(alpha >-1.)</span> As an application, we discuss the boundedness of composition operators on <span>({mathcal {D}}_{alpha }(Pi ^{+}))</span> induced by linear fractional self-maps of <span>(Pi ^{+}.)</span> Second, we characterize composition operators and their adjoints induced by affine self-maps of <span>(Pi ^{+})</span> that have universal translates on <span>({mathcal {D}}_{alpha }(Pi ^{+}).)</span> Moreover, we investigate which composition operators and their adjoints induced by hyperbolic non-automorphism self-maps of the open unit disk <span>({mathbb {D}})</span> have universal translates on weighted Dirichlet spaces <span>({mathcal {D}}_{alpha }({mathbb {D}}))</span> for <span>(alpha >-1.)</span> Finally, we consider the minimal invariant subspaces of the composition operators that have universal translates.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-31DOI: 10.1007/s43037-023-00312-y
Jinxun Wang, Tao Qian
{"title":"Orthogonalization in Clifford Hilbert modules and applications","authors":"Jinxun Wang, Tao Qian","doi":"10.1007/s43037-023-00312-y","DOIUrl":"https://doi.org/10.1007/s43037-023-00312-y","url":null,"abstract":"","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135869790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-30DOI: 10.1007/s43037-023-00311-z
Yaxu Li
{"title":"Iterative kernel density estimation from noisy-dependent observations","authors":"Yaxu Li","doi":"10.1007/s43037-023-00311-z","DOIUrl":"https://doi.org/10.1007/s43037-023-00311-z","url":null,"abstract":"","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136106012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1007/s43037-023-00309-7
Giorgia Bellomonte, Camillo Trapani
Abstract Quasi *-algebras possessing a sufficient family $$mathcal {M}$$ M of invariant positive sesquilinear forms carry several topologies related to $$mathcal {M}$$ M which make every *-representation continuous. This leads to define the class of locally convex quasi GA*-algebras whose main feature consists in the fact that the family of their bounded elements, with respect to the family $$mathcal {M}$$ M , is a dense C*-algebra.
{"title":"Topological aspects of quasi *-algebras with sufficiently many *-representations","authors":"Giorgia Bellomonte, Camillo Trapani","doi":"10.1007/s43037-023-00309-7","DOIUrl":"https://doi.org/10.1007/s43037-023-00309-7","url":null,"abstract":"Abstract Quasi *-algebras possessing a sufficient family $$mathcal {M}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>M</mml:mi> </mml:math> of invariant positive sesquilinear forms carry several topologies related to $$mathcal {M}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>M</mml:mi> </mml:math> which make every *-representation continuous. This leads to define the class of locally convex quasi GA*-algebras whose main feature consists in the fact that the family of their bounded elements, with respect to the family $$mathcal {M}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>M</mml:mi> </mml:math> , is a dense C*-algebra.","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135606998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1007/s43037-023-00304-y
Fabrizio Colombo, Antonino De Martino, Stefano Pinton
Abstract Harmonic and polyanalytic functional calculi have been recently defined for bounded commuting operators. Their definitions are based on the Cauchy formula of slice hyperholomorphic functions and on the factorization of the Laplace operator in terms of the Cauchy–Fueter operator $${mathcal{D}}$$ D and of its conjugate $$overline{{mathcal{D}}}.$$ D¯. Thanks to the Fueter extension theorem, when we apply the operator $${mathcal{D}}$$ D to slice hyperholomorphic functions, we obtain harmonic functions and via the Cauchy formula of slice hyperholomorphic functions, we establish an integral representation for harmonic functions. This integral formula is used to define the harmonic functional calculus on the S -spectrum. Another possibility is to apply the conjugate of the Cauchy–Fueter operator to slice hyperholomorphic functions. In this case, with a similar procedure we obtain the class of polyanalytic functions, their integral representation, and the associated polyanalytic functional calculus. The aim of this paper is to extend the harmonic and the polyanalytic functional calculi to the case of unbounded operators and to prove some of the most important properties. These two functional calculi belong to so called fine structures on the S -spectrum in the quaternionic setting. Fine structures on the S -spectrum associated with Clifford algebras constitute a new research area that deeply connects different research fields such as operator theory, harmonic analysis, and hypercomplex analysis.
{"title":"Harmonic and polyanalytic functional calculi on the S-spectrum for unbounded operators","authors":"Fabrizio Colombo, Antonino De Martino, Stefano Pinton","doi":"10.1007/s43037-023-00304-y","DOIUrl":"https://doi.org/10.1007/s43037-023-00304-y","url":null,"abstract":"Abstract Harmonic and polyanalytic functional calculi have been recently defined for bounded commuting operators. Their definitions are based on the Cauchy formula of slice hyperholomorphic functions and on the factorization of the Laplace operator in terms of the Cauchy–Fueter operator $${mathcal{D}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>D</mml:mi> </mml:math> and of its conjugate $$overline{{mathcal{D}}}.$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Thanks to the Fueter extension theorem, when we apply the operator $${mathcal{D}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>D</mml:mi> </mml:math> to slice hyperholomorphic functions, we obtain harmonic functions and via the Cauchy formula of slice hyperholomorphic functions, we establish an integral representation for harmonic functions. This integral formula is used to define the harmonic functional calculus on the S -spectrum. Another possibility is to apply the conjugate of the Cauchy–Fueter operator to slice hyperholomorphic functions. In this case, with a similar procedure we obtain the class of polyanalytic functions, their integral representation, and the associated polyanalytic functional calculus. The aim of this paper is to extend the harmonic and the polyanalytic functional calculi to the case of unbounded operators and to prove some of the most important properties. These two functional calculi belong to so called fine structures on the S -spectrum in the quaternionic setting. Fine structures on the S -spectrum associated with Clifford algebras constitute a new research area that deeply connects different research fields such as operator theory, harmonic analysis, and hypercomplex analysis.","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135761239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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