Abstract Given Banach space operators Ai, Bi (i = 1, 2), let δi denote (the generalised derivation) δi(X) = (LAi − RBi )(X) = AiX − XBi. If 0 ∈ σa(Bi), i = 1, 2, and if Δδ1,δ2n(I)=(Lδ1Rδ1-I)n(I)=0 Delta _{{delta _1},delta 2}^nleft( I right) = {left( {{L_{{delta _1}}}{R_{{delta _1}}} - I} right)^n}left( I right) = 0 , then ΔA1,A2n(I)=0 Delta _{{A_1},A2}^nleft( I right) = 0 . For Hilbert space pairs (A, B) such that 0 ∈ σa(B*) and Δδ*,δn(I)=0(i.e., δ is n-isometric) Delta _{{delta ^*},delta }^nleft( I right) = 0left( {i.e.,,delta ,is,n - isometric} right) , where δ= δA,B and δ* = δA* ,B*, this implies ΔA*,An(I)=0 Delta _{{A^*},A}^nleft( I right) = 0 (and hence there exists a positivie integer m ≤ n such that A is strictly m-isometric). If Δδ*,δn(I)=0 Delta _{{delta ^*},delta }^nleft( I right) = 0 , then there exists a scalar λ such that 0 ∈ σa((B − λI)*) and, given δ is strictly n-isometric, there exists a positive integer m ≤ n such that A − λI is strictly m-isometric. Furthermore, there exist decompositions ℋ = ℋ1 ⊕ ℋ2 and ℋ = ℋ11 ⊕ ℋ22 of ℋ and ti-nilpotent operators Ni (i = 1, 2) such that either A − λI = αI + N1 and B − λI = (0I|ℋ1 ⊕ 2eit I|ℋ2 ) + N2, or, A − λI = αI + N1, α = eit, 0 ≤ t < 2π, and B − λI = (0I|ℋ11 ⊕ 2eit I|ℋ22 ) + N2, or, A − λ = (α1I|ℋ1 ⊕ α2I|ℋ2 ) + N1 and Bλ= (0I|ℋ11 ⊕ μI|ℋ22 ) + N2, where μ = eit|μ|, 0 ≤ t < 2π, 0 < |μ| < 2, α1=eit| μ |+i4-| μ |22 {alpha _1} = {e^{it}}{{left| mu right| + isqrt {4 - {{left| mu right|}^2}} } over 2} and α2=eit| μ |-i4-| μ |22 {alpha _2} = {e^{it}}{{left| mu right| - isqrt {4 - {{left| mu right|}^2}} } over 2} .
给定Banach空间算子Ai, Bi (i = 1,2)，令δi表示(广义导数)δi(X) = (LAi−RBi)(X) = AiX−XBi。如果0∈σa(Bi)， i = 1,2，如果Δδ1，δ2n(i) =(Lδ1Rδ1-I)n(i) =0 Delta ＿{{delta ＿1}，delta 2}^nleft(1) right) = {left( {{我……{{delta ＿1}}}{r＿{{delta ＿1}}} -我} right)^n}left(1) right)=0，则ΔA1,A2n(I)=0 Delta ＿{{a＿1}， a}^nleft(1) right) = 0。对于希尔伯特空间对(A, B)，使得0∈σa(B*)和Δδ*，δn(I)=0(即δ是n等距) Delta ＿{{delta ^*}，delta }^nleft(1) right) = 0left( {即，，delta ，是，n -等距} right)，其中δ= δ a,B和δ* = δ a *， B*，这意味着ΔA*，An(I)=0 Delta ＿{{a ^*}， a}^nleft(1) right) = 0(因此存在一个正整数m≤n，使得a是严格m等距的)。如果Δδ*，δn(I)=0 Delta ＿{{delta ^*}，delta }^nleft(1) right) = 0，则存在一个标量λ使得0∈σa((B−λ i)*)，并且，给定δ是严格n等距的，则存在一个正整数m≤n使得a−λ i是严格m等距的。进一步地，存在着分解h = h = h 1⊕h 2和h = h 11⊕h 22的h和反幂零算子Ni (i = 1,2)，使得A−λ i = α i + N1和B−λ i = (0I| h 1⊕2eit i | h 2) + N2，或者A−λ= (α i + N1， α =eit, 0≤t < 2π， B−λ i = (0I| h 11⊕2eit i | h 22) + N2，或者A−λ= (α 1i | h 1⊕α 2i | h 2) + N1和Bλ= (0I| h 11⊕μ i | h 22) + N2，其中μ =eit| μ|， 0≤t < 2π， 0 < |μ| < 2， α1=eit| μ| +i4-| μ| 22 {alpha ＿1} ＝ {e^{它}}{{left| mu right| + Isqrt {4 - {{left| mu right|}^2}} } over 2} α2=eit| μ |-i4-| μ |22 {alpha ＿2} ＝ {e^{它}}{{left| mu right| - Isqrt {4 - {{left| mu right|}^2}} } over 2} ．
{"title":"m-isometric generalised derivations","authors":"B. Duggal, I. H. Kim","doi":"10.1515/conop-2022-0135","DOIUrl":"https://doi.org/10.1515/conop-2022-0135","url":null,"abstract":"Abstract Given Banach space operators Ai, Bi (i = 1, 2), let δi denote (the generalised derivation) δi(X) = (LAi − RBi )(X) = AiX − XBi. If 0 ∈ σa(Bi), i = 1, 2, and if Δδ1,δ2n(I)=(Lδ1Rδ1-I)n(I)=0 Delta _{{delta _1},delta 2}^nleft( I right) = {left( {{L_{{delta _1}}}{R_{{delta _1}}} - I} right)^n}left( I right) = 0 , then ΔA1,A2n(I)=0 Delta _{{A_1},A2}^nleft( I right) = 0 . For Hilbert space pairs (A, B) such that 0 ∈ σa(B*) and Δδ*,δn(I)=0(i.e., δ is n-isometric) Delta _{{delta ^*},delta }^nleft( I right) = 0left( {i.e.,,delta ,is,n - isometric} right) , where δ= δA,B and δ* = δA* ,B*, this implies ΔA*,An(I)=0 Delta _{{A^*},A}^nleft( I right) = 0 (and hence there exists a positivie integer m ≤ n such that A is strictly m-isometric). If Δδ*,δn(I)=0 Delta _{{delta ^*},delta }^nleft( I right) = 0 , then there exists a scalar λ such that 0 ∈ σa((B − λI)*) and, given δ is strictly n-isometric, there exists a positive integer m ≤ n such that A − λI is strictly m-isometric. Furthermore, there exist decompositions ℋ = ℋ1 ⊕ ℋ2 and ℋ = ℋ11 ⊕ ℋ22 of ℋ and ti-nilpotent operators Ni (i = 1, 2) such that either A − λI = αI + N1 and B − λI = (0I|ℋ1 ⊕ 2eit I|ℋ2 ) + N2, or, A − λI = αI + N1, α = eit, 0 ≤ t < 2π, and B − λI = (0I|ℋ11 ⊕ 2eit I|ℋ22 ) + N2, or, A − λ = (α1I|ℋ1 ⊕ α2I|ℋ2 ) + N1 and Bλ= (0I|ℋ11 ⊕ μI|ℋ22 ) + N2, where μ = eit|μ|, 0 ≤ t < 2π, 0 < |μ| < 2, α1=eit| μ |+i4-| μ |22 {alpha _1} = {e^{it}}{{left| mu right| + isqrt {4 - {{left| mu right|}^2}} } over 2} and α2=eit| μ |-i4-| μ |22 {alpha _2} = {e^{it}}{{left| mu right| - isqrt {4 - {{left| mu right|}^2}} } over 2} .","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"139 - 150"},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45651276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The aim of this paper is to carry out an analysis of five trascendental operators acting on the space of slice regular functions, namely *-exponential, *-sine and *-cosine and their hyperbolic analogues. The first three of them were introduced by Colombo, Sabadini and Struppa and some features of *-exponential were investigated in a previous paper by Altavilla and the author. We show how exp*(f ), sin*(f ), cos*(f ), sinh*(f ) and cosh*(f ) can be written in terms of the real and the vector part of the function f and we examine the relation between cos* and cosh* when the domain Ω is product and when it is slice. In particular we prove that when Ω is slice, then cos*(f ) = cosh*(f * I) holds if and only if f is ℂI preserving, while in the case Ω is product there is a much larger family of slice regular functions for which the above relation holds.
{"title":"Transcendental operators acting on slice regular functions","authors":"C. de Fabritiis","doi":"10.1515/conop-2022-0002","DOIUrl":"https://doi.org/10.1515/conop-2022-0002","url":null,"abstract":"Abstract The aim of this paper is to carry out an analysis of five trascendental operators acting on the space of slice regular functions, namely *-exponential, *-sine and *-cosine and their hyperbolic analogues. The first three of them were introduced by Colombo, Sabadini and Struppa and some features of *-exponential were investigated in a previous paper by Altavilla and the author. We show how exp*(f ), sin*(f ), cos*(f ), sinh*(f ) and cosh*(f ) can be written in terms of the real and the vector part of the function f and we examine the relation between cos* and cosh* when the domain Ω is product and when it is slice. In particular we prove that when Ω is slice, then cos*(f ) = cosh*(f * I) holds if and only if f is ℂI preserving, while in the case Ω is product there is a much larger family of slice regular functions for which the above relation holds.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"6 - 18"},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48278181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Valentin V. Andreev, Miron B. Bekker, Joseph A. Cima
Abstract In this article we continue our investigation of the Paatero space. We prove that the intersection of every Paatero class V(k) with every Hardy space Hp is closed in that Hp and associate singular continuous measures to elements of V(k). There have been no examples in the literature of functions in V(k) with zeros in the unit disk other than the one at the origin. We close this gap in the literature. We derive a representation of the measure associated to a function in V(k) for functions whose derivatives are rational, or algebraic, or transcendental functions in the unit disk.Finally, we consider the notion of regulated domains, introduced by Pommerenke and show that there are regulated domains whose boundary is not of bounded boundary rotation.
{"title":"Paatero’s V(k) space II","authors":"Valentin V. Andreev, Miron B. Bekker, Joseph A. Cima","doi":"10.1515/conop-2022-0134","DOIUrl":"https://doi.org/10.1515/conop-2022-0134","url":null,"abstract":"Abstract In this article we continue our investigation of the Paatero space. We prove that the intersection of every Paatero class V(k) with every Hardy space Hp is closed in that Hp and associate singular continuous measures to elements of V(k). There have been no examples in the literature of functions in V(k) with zeros in the unit disk other than the one at the origin. We close this gap in the literature. We derive a representation of the measure associated to a function in V(k) for functions whose derivatives are rational, or algebraic, or transcendental functions in the unit disk.Finally, we consider the notion of regulated domains, introduced by Pommerenke and show that there are regulated domains whose boundary is not of bounded boundary rotation.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"151 - 159"},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48196418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper explores the long journey from projective tensor products of a pair of Banach spaces, passing through the definition of nuclear operators still on the realm of projective tensor products, to the of notion of trace-class operators on a Hilbert space, and shows how and why these concepts (nuclear and trace-class operators, that is) agree in the end.
{"title":"Trace-Class and Nuclear Operators","authors":"C. Kubrusly","doi":"10.1515/conop-2022-0127","DOIUrl":"https://doi.org/10.1515/conop-2022-0127","url":null,"abstract":"Abstract This paper explores the long journey from projective tensor products of a pair of Banach spaces, passing through the definition of nuclear operators still on the realm of projective tensor products, to the of notion of trace-class operators on a Hilbert space, and shows how and why these concepts (nuclear and trace-class operators, that is) agree in the end.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"25 3","pages":"53 - 69"},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41294414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we improve some numerical radius inequalities for Hilbert space operators under suitable condition. We also compare our results with some known results. As application of our result, we obtain an operator inequality.
{"title":"Refinements of numerical radius inequalities using the Kantorovich ratio","authors":"Elham Nikzat, M. Omidvar","doi":"10.1515/conop-2022-0128","DOIUrl":"https://doi.org/10.1515/conop-2022-0128","url":null,"abstract":"Abstract In this paper, we improve some numerical radius inequalities for Hilbert space operators under suitable condition. We also compare our results with some known results. As application of our result, we obtain an operator inequality.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"70 - 74"},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43090141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Over the past several years, optimal polynomial approximants (OPAs) have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the properties of their zeros. In this paper, we introduce the notion of optimal polynomial approximant in the space Lp, 1 ≤ p ≤ ∞. We begin our treatment by showing existence and uniqueness for 1 < p < ∞. For the extreme cases of p = 1 and p = ∞, we show that uniqueness does not necessarily hold. We continue our development by elaborating on the special case of L2. Here, we create a test to determine whether or not a given 1st degree OPA is zero-free in ̄𝔻. Afterward, we shed light on an orthogonality condition in Lp. This allows us to study OPAs in Lp with the additional tools from the L2 setting. Throughout this paper, we focus many of our discussions on the zeros of OPAs. In particular, we show that if 1 < p < ∞, f ∈ Hp, and f(0) ≠ 0, then there exists a disk, centered at the origin, in which all the associated OPAs are zero-free. Toward the end of this paper, we use the orthogonality condition to compute the coefficients of some OPAs in Lp. To inspire further research in the general theory, we pose several open questions throughout our discussions.
在过去的几年里，最优多项式近似(OPAs)在许多不同的函数空间中得到了研究。在这种情况下，许多论文都致力于研究它们的零的性质。本文引入了空间Lp, 1≤p≤∞上最优多项式近似的概念。我们首先证明1 < p <∞的存在唯一性。对于p = 1和p =∞的极端情况，我们证明唯一性不一定成立。我们通过阐述L2的特殊情况来继续我们的发展。在这里，我们创建一个测试来确定给定的一阶OPA是否为零。然后，我们阐明了Lp中的一个正交性条件。这使我们能够使用L2设置的附加工具研究Lp中的opa。在本文中，我们将重点讨论opa的零点。特别地，我们证明了如果1 < p <∞，f∈Hp，且f(0)≠0，那么存在一个以原点为中心的圆盘，其中所有相关的opa都是零自由的。在本文的最后，我们利用正交性条件计算了Lp中一些opa的系数。为了启发一般理论的进一步研究，我们在讨论中提出了几个开放性问题。
{"title":"Optimal Polynomial Approximants in Lp","authors":"R. Centner","doi":"10.1515/conop-2022-0131","DOIUrl":"https://doi.org/10.1515/conop-2022-0131","url":null,"abstract":"Abstract Over the past several years, optimal polynomial approximants (OPAs) have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the properties of their zeros. In this paper, we introduce the notion of optimal polynomial approximant in the space Lp, 1 ≤ p ≤ ∞. We begin our treatment by showing existence and uniqueness for 1 < p < ∞. For the extreme cases of p = 1 and p = ∞, we show that uniqueness does not necessarily hold. We continue our development by elaborating on the special case of L2. Here, we create a test to determine whether or not a given 1st degree OPA is zero-free in ̄𝔻. Afterward, we shed light on an orthogonality condition in Lp. This allows us to study OPAs in Lp with the additional tools from the L2 setting. Throughout this paper, we focus many of our discussions on the zeros of OPAs. In particular, we show that if 1 < p < ∞, f ∈ Hp, and f(0) ≠ 0, then there exists a disk, centered at the origin, in which all the associated OPAs are zero-free. Toward the end of this paper, we use the orthogonality condition to compute the coefficients of some OPAs in Lp. To inspire further research in the general theory, we pose several open questions throughout our discussions.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"96 - 113"},"PeriodicalIF":0.6,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43090006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jessica Doctor, T. Hodges, Scott R. Kaschner, Alexander McFarland, D. Thompson
Abstract Previously, spectra of certain weighted composition operators W ѱ, φ on H2 were determined under one of two hypotheses: either φ converges under iteration to the Denjoy-Wolff point uniformly on all of 𝔻 rather than simply on compact subsets, or φ is “essentially linear fractional.” We show that if φ is a quadratic self-map of 𝔻 of parabolic type, then the spectrum of Wѱ, φ can be found when these maps exhibit both of the aforementioned properties, and we determine which symbols do so.
{"title":"Spectra of Weighted Composition Operators with Quadratic Symbols","authors":"Jessica Doctor, T. Hodges, Scott R. Kaschner, Alexander McFarland, D. Thompson","doi":"10.1515/conop-2022-0129","DOIUrl":"https://doi.org/10.1515/conop-2022-0129","url":null,"abstract":"Abstract Previously, spectra of certain weighted composition operators W ѱ, φ on H2 were determined under one of two hypotheses: either φ converges under iteration to the Denjoy-Wolff point uniformly on all of 𝔻 rather than simply on compact subsets, or φ is “essentially linear fractional.” We show that if φ is a quadratic self-map of 𝔻 of parabolic type, then the spectrum of Wѱ, φ can be found when these maps exhibit both of the aforementioned properties, and we determine which symbols do so.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"75 - 85"},"PeriodicalIF":0.6,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46660238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract For p ∈ (1, ∞) {2}, some properties of the space ℳp of multipliers on ℓpA are derived. In particular, the failure of the weak parallelogram laws and the Pythagorean inequalities is demonstrated for ℳp. It is also shown that the extremal multipliers on the ℓpA spaces are exactly the monomials, in stark contrast to the p = 2 case.
{"title":"On the geometry of the multiplier space of ℓpA","authors":"Christopher Felder, R. Cheng","doi":"10.1515/conop-2022-0126","DOIUrl":"https://doi.org/10.1515/conop-2022-0126","url":null,"abstract":"Abstract For p ∈ (1, ∞) {2}, some properties of the space ℳp of multipliers on ℓpA are derived. In particular, the failure of the weak parallelogram laws and the Pythagorean inequalities is demonstrated for ℳp. It is also shown that the extremal multipliers on the ℓpA spaces are exactly the monomials, in stark contrast to the p = 2 case.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"41 - 52"},"PeriodicalIF":0.6,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41705437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this survey, we present several results related to characterizing the surjective isometries on Banach sequence spaces. Our survey includes full proofs of these characterizations for the classical spaces as well as more recent results for combinatorial Banach spaces and Tsirelson-type spaces. Along the way, we pose many open problems related to the structure of the group of surjective isometries for various Banach spaces.
{"title":"Surjective isometries on Banach sequence spaces: A survey","authors":"Leandro Antunes, K. Beanland","doi":"10.1515/conop-2022-0125","DOIUrl":"https://doi.org/10.1515/conop-2022-0125","url":null,"abstract":"Abstract In this survey, we present several results related to characterizing the surjective isometries on Banach sequence spaces. Our survey includes full proofs of these characterizations for the classical spaces as well as more recent results for combinatorial Banach spaces and Tsirelson-type spaces. Along the way, we pose many open problems related to the structure of the group of surjective isometries for various Banach spaces.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"19 - 40"},"PeriodicalIF":0.6,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43901705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂn cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n. The proof is based on the unitary similarity of A to a compressed shift operator SB generated by a finite Blaschke product B. We then use the description of the numerical range of SB as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions.
{"title":"On partial isometries with circular numerical range","authors":"E. Wegert, I. Spitkovsky","doi":"10.1515/conop-2020-0121","DOIUrl":"https://doi.org/10.1515/conop-2020-0121","url":null,"abstract":"Abstract In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂn cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n. The proof is based on the unitary similarity of A to a compressed shift operator SB generated by a finite Blaschke product B. We then use the description of the numerical range of SB as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"176 - 186"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45378147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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