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":"15 1","pages":"0"},"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}
Pub Date : 2023-09-26DOI: 10.1007/s43037-023-00303-z
Rewayat Khan, Ji Eun Lee
{"title":"Characterizations of matrix-valued asymmetric truncated Hankel operators","authors":"Rewayat Khan, Ji Eun Lee","doi":"10.1007/s43037-023-00303-z","DOIUrl":"https://doi.org/10.1007/s43037-023-00303-z","url":null,"abstract":"","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134960414","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 : 2023-09-15DOI: 10.1007/s43037-023-00298-7
Xiao-Li Zhang, Yun-Zhang Li
{"title":"Publisher Correction: Quaternionic Gabor frame characterization and the density theorem","authors":"Xiao-Li Zhang, Yun-Zhang Li","doi":"10.1007/s43037-023-00298-7","DOIUrl":"https://doi.org/10.1007/s43037-023-00298-7","url":null,"abstract":"","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135396204","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 : 2023-09-12DOI: 10.1007/s43037-023-00300-2
Davit Baramidze, Lasha Baramidze, Lars-Erik Perssson, George Tephnadze
Abstract In this paper, we derive the maximal subspace of natural numbers $$left{ n_{k}:kge 0right} ,$$ nk:k≥0, such that the restricted maximal operator, defined by $${sup }_{kin {mathbb {N}}}left| sigma _{n_{k}}Fright| $$ supk∈NσnkF on this subspace of Fejér means of Walsh–Fourier series is bounded from the martingale Hardy space $$H_{1/2}$$ H1/2 to the Lebesgue space $$L_{1/2}.$$ L1/2. The sharpness of this result is also proved.
摘要本文导出了自然数的极大子空间$$left{ n_{k}:kge 0right} ,$$ n k: k≥0，使得约束极大算子(由$${sup }_{kin {mathbb {N}}}left| sigma _{n_{k}}Fright| $$ sup k∈n σ n k F定义)在这个w -傅里叶级数的fej均值子空间上由鞅Hardy空间$$H_{1/2}$$ H 1 / 2有界到Lebesgue空间$$L_{1/2}.$$ L 1 / 2。并证明了该结果的清晰性。
{"title":"Some new restricted maximal operators of Fejér means of Walsh–Fourier series","authors":"Davit Baramidze, Lasha Baramidze, Lars-Erik Perssson, George Tephnadze","doi":"10.1007/s43037-023-00300-2","DOIUrl":"https://doi.org/10.1007/s43037-023-00300-2","url":null,"abstract":"Abstract In this paper, we derive the maximal subspace of natural numbers $$left{ n_{k}:kge 0right} ,$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mfenced> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> such that the restricted maximal operator, defined by $${sup }_{kin {mathbb {N}}}left| sigma _{n_{k}}Fright| $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mo>sup</mml:mo> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:mfenced> <mml:msub> <mml:mi>σ</mml:mi> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:msub> <mml:mi>F</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> on this subspace of Fejér means of Walsh–Fourier series is bounded from the martingale Hardy space $$H_{1/2}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> to the Lebesgue space $$L_{1/2}.$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> The sharpness of this result is also proved.","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135825983","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 : 2023-09-04DOI: 10.1007/s43037-023-00295-w
Alejandro Miralles
{"title":"Bounded below composition operators on the space of Bloch functions on the unit ball of a Hilbert space","authors":"Alejandro Miralles","doi":"10.1007/s43037-023-00295-w","DOIUrl":"https://doi.org/10.1007/s43037-023-00295-w","url":null,"abstract":"","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43658863","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 : 2023-09-01DOI: 10.1007/s43037-023-00299-6
J. Bračič
{"title":"Reflexivity of finite-dimensional sets of operators","authors":"J. Bračič","doi":"10.1007/s43037-023-00299-6","DOIUrl":"https://doi.org/10.1007/s43037-023-00299-6","url":null,"abstract":"","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46833441","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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