Pub Date : 2019-03-13DOI: 10.7900/jot.2019mar23.2264
Takahiro Hasebe, Yuki Ueda
We study unimodality for free multiplicative convolution with free normal distributions {λt}t>0 on the unit circle. We give four results on unimodality for μ⊠λt: (1) if μ is a symmetric unimodal distribution on the unit circle then so is μ⊠λt at any time t>0; (2) if μ is a symmetric distribution on T supported on {eiθ:θ∈[−φ,φ]} for some φ∈(0,π2), then μ⊠λt is unimodal for sufficiently large t>0; (3) b⊠λt is not unimodal at any time t>0, where b is the equally weighted Bernoulli distribution on {1,−1}; (4) λt is not freely strongly unimodal for sufficiently small t>0. Moreover, we study unimodality for classical multiplicative convolution, which is useful in proving the above four results.
{"title":"Unimodality for free multiplicative convolution with free normal distributions on the unit circle","authors":"Takahiro Hasebe, Yuki Ueda","doi":"10.7900/jot.2019mar23.2264","DOIUrl":"https://doi.org/10.7900/jot.2019mar23.2264","url":null,"abstract":"We study unimodality for free multiplicative convolution with free normal distributions {λt}t>0 on the unit circle. We give four results on unimodality for μ⊠λt: (1) if μ is a symmetric unimodal distribution on the unit circle then so is μ⊠λt at any time t>0; (2) if μ is a symmetric distribution on T supported on {eiθ:θ∈[−φ,φ]} for some φ∈(0,π2), then μ⊠λt is unimodal for sufficiently large t>0; (3) b⊠λt is not unimodal at any time t>0, where b is the equally weighted Bernoulli distribution on {1,−1}; (4) λt is not freely strongly unimodal for sufficiently small t>0. Moreover, we study unimodality for classical multiplicative convolution, which is useful in proving the above four results.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48744564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-03-12DOI: 10.7900/jot.2019dec19.2300
V. Alekseev, Rahel Brugger
We show a rigidity result for subfactors that are normalized by a representation of a lattice Γ in a higher rank simple Lie group with trivial center into a finite factor. This implies that every subfactor of LΓ which is normalized by the natural copy of Γ is trivial or of finite index.
{"title":"A rigidity result for normalized subfactors","authors":"V. Alekseev, Rahel Brugger","doi":"10.7900/jot.2019dec19.2300","DOIUrl":"https://doi.org/10.7900/jot.2019dec19.2300","url":null,"abstract":"We show a rigidity result for subfactors that are normalized by a representation of a lattice Γ in a higher rank simple Lie group with trivial center into a finite factor. This implies that every subfactor of LΓ which is normalized by the natural copy of Γ is trivial or of finite index.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45701058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-03-05DOI: 10.7900/jot.2020feb05.2298
C. Badea, Laurian Suciu
A continuous linear Hilbert space operator S is said to be a 2-isometry if the operator S and its adjoint S∗ satisfy the relation S∗2S2−2S∗S+I=0. We study operators having liftings or dilations to 2-isometries. The adjoint of an operator which admits such liftings is the restriction of a backward shift on a Hilbert space of vector-valued analytic functions. These results are applied to concave operators and to operators similar to contractions. Two types of liftings to 2-isometries, as well as the extensions induced by them, are constructed and isomorphic minimal liftings are discussed.
{"title":"Hilbert space operators with two-isometric dilations","authors":"C. Badea, Laurian Suciu","doi":"10.7900/jot.2020feb05.2298","DOIUrl":"https://doi.org/10.7900/jot.2020feb05.2298","url":null,"abstract":"A continuous linear Hilbert space operator S is said to be a 2-isometry if the operator S and its adjoint S∗ satisfy the relation S∗2S2−2S∗S+I=0. We study operators having liftings or dilations to 2-isometries. The adjoint of an operator which admits such liftings is the restriction of a backward shift on a Hilbert space of vector-valued analytic functions. These results are applied to concave operators and to operators similar to contractions. Two types of liftings to 2-isometries, as well as the extensions induced by them, are constructed and isomorphic minimal liftings are discussed.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44285381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-03-05DOI: 10.7900/jot.2018may14.2238
Chi-Kwong Li, M. Tsai, Ya-Shu Wang, N. Wong
Let Mm,n be the space of m×n real or complex rectangular matrices. Two matrices A,B∈Mm,n are disjoint if A∗B=0n and AB∗=0m. We show that a linear map Φ:Mm,n→Mr,s preserving disjointness exactly when Φ(A)=U⎛⎜⎝A⊗Q1000At⊗Q2000⎞⎟⎠V,∀A∈Mm,n, for some unitary matrices U∈Mr,r and V∈Ms,s, and positive diagonal matrices Q1,Q2, where Q1 or Q2 may be vacuous. The result is used to characterize nonsurjective linear maps between rectangular matrix spaces preserving (zero) JB∗-triple products, the Schatten p-norms or the Ky--Fan k-norms.
{"title":"Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms","authors":"Chi-Kwong Li, M. Tsai, Ya-Shu Wang, N. Wong","doi":"10.7900/jot.2018may14.2238","DOIUrl":"https://doi.org/10.7900/jot.2018may14.2238","url":null,"abstract":"Let Mm,n be the space of m×n real or complex rectangular matrices. Two matrices A,B∈Mm,n are disjoint if A∗B=0n and AB∗=0m. We show that a linear map Φ:Mm,n→Mr,s preserving disjointness exactly when Φ(A)=U⎛⎜⎝A⊗Q1000At⊗Q2000⎞⎟⎠V,∀A∈Mm,n, for some unitary matrices U∈Mr,r and V∈Ms,s, and positive diagonal matrices Q1,Q2, where Q1 or Q2 may be vacuous. The result is used to characterize nonsurjective linear maps between rectangular matrix spaces preserving (zero) JB∗-triple products, the Schatten p-norms or the Ky--Fan k-norms.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43062237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-02-06DOI: 10.7900/jot.2019aug29.2271
I. Charlesworth, Brent Nelson
We introduce a free probabilistic quantity called free Stein irregularity, which is defined in terms of free Stein discrepancies. It turns out that this quantity is related via a simple formula to the Murray-von Neumann dimension of the closure of the domain of the adjoint of the non-commutative Jacobian associated to Voiculescu's free difference quotients. We call this dimension the free Stein dimension, and show that it is a ∗-algebra invariant. We relate these quantities to the free Fisher information, the non-microstates free entropy, and the non-microstates free entropy dimension. In the one-variable case, we show that the free Stein dimension agrees with the free entropy dimension, and in the multivariable case compute it in a number of examples.
{"title":"Free Stein irregularity and dimension","authors":"I. Charlesworth, Brent Nelson","doi":"10.7900/jot.2019aug29.2271","DOIUrl":"https://doi.org/10.7900/jot.2019aug29.2271","url":null,"abstract":"We introduce a free probabilistic quantity called free Stein irregularity, which is defined in terms of free Stein discrepancies. It turns out that this quantity is related via a simple formula to the Murray-von Neumann dimension of the closure of the domain of the adjoint of the non-commutative Jacobian associated to Voiculescu's free difference quotients. We call this dimension the free Stein dimension, and show that it is a ∗-algebra invariant. We relate these quantities to the free Fisher information, the non-microstates free entropy, and the non-microstates free entropy dimension. In the one-variable case, we show that the free Stein dimension agrees with the free entropy dimension, and in the multivariable case compute it in a number of examples.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45163703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-20DOI: 10.7900/JOT.2018NOV21.2215
V. Crismale, Y. Lu
We investigate the spectrum for partial sums of m position (or gaussian) operators on monotone Fock space based on ℓ2(N). In the basic case of the first consecutive operators, we prove it coincides with the support of the vacuum distribution. Thus, the right endpoint of the support gives the norm. In the general case, we get that the last property for norm still holds. As any single position operator has the vacuum symmetric Bernoulli law, and the whole of them is a monotone independent family of random variables, the vacuum distribution for partial sums of n operators can be seen as the monotone binomial with n trials. It is a discrete measure supported on a finite set, and we exhibit recurrence formulas to compute its atoms and probability function as well. Moreover, lower and upper bounds for the right endpoints of the supports are given.
{"title":"Vacuum distribution, norm and spectral properties for sums of monotone position operators","authors":"V. Crismale, Y. Lu","doi":"10.7900/JOT.2018NOV21.2215","DOIUrl":"https://doi.org/10.7900/JOT.2018NOV21.2215","url":null,"abstract":"We investigate the spectrum for partial sums of m position (or gaussian) operators on monotone Fock space based on ℓ2(N). In the basic case of the first consecutive operators, we prove it coincides with the support of the vacuum distribution. Thus, the right endpoint of the support gives the norm. In the general case, we get that the last property for norm still holds. As any single position operator has the vacuum symmetric Bernoulli law, and the whole of them is a monotone independent family of random variables, the vacuum distribution for partial sums of n operators can be seen as the monotone binomial with n trials. It is a discrete measure supported on a finite set, and we exhibit recurrence formulas to compute its atoms and probability function as well. Moreover, lower and upper bounds for the right endpoints of the supports are given.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49494301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-15DOI: 10.7900/jot.2017dec22.2183
P. Kurasov, Annemarie Luger, Christoph Neuner
In this paper self-adjoint realizations of the formal expression Aα:=A+α⟨ϕ,⋅⟩ϕ are described, where α∈R∪{∞}, the operator A is self-adjoint in a Hilbert space H and ϕ is a supersingular element from the scale space H−n−2(A)∖H−n−1(A) for n⩾1. The crucial point is that the spectrum of A may consist of the whole real line. We construct two models to describe the family (Aα). It can be interpreted in a Hilbert space with a twisted version of Krein's formula, or with a more classical version of Krein's formula but in a Pontryagin space. Finally, we compare the two approaches in terms of the respective Q-functions.
{"title":"On supersingular perturbations of non-semibounded self-adjoint operators","authors":"P. Kurasov, Annemarie Luger, Christoph Neuner","doi":"10.7900/jot.2017dec22.2183","DOIUrl":"https://doi.org/10.7900/jot.2017dec22.2183","url":null,"abstract":"In this paper self-adjoint realizations of the formal expression Aα:=A+α⟨ϕ,⋅⟩ϕ are described, where α∈R∪{∞}, the operator A is self-adjoint in a Hilbert space H and ϕ is a supersingular element from the scale space H−n−2(A)∖H−n−1(A) for n⩾1. The crucial point is that the spectrum of A may consist of the whole real line. We construct two models to describe the family (Aα). It can be interpreted in a Hilbert space with a twisted version of Krein's formula, or with a more classical version of Krein's formula but in a Pontryagin space. Finally, we compare the two approaches in terms of the respective Q-functions.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44237168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-15DOI: 10.7900/jot.2017dec06.2200
Xiaoyan Zhou, Junsheng Fang
Let M be a finite von Neumann algebra (respectively, a type II1 factor) and let N⊂M be a II1 factor (respectively, N⊂M have an atomic part). We prove that if the inclusion N⊂M is amenable, then implies the identity map on M has an approximate factorization through Mm(C)⊗N via trace preserving normal unital completely positive maps, which is a generalization of a result of Haagerup. We also prove two permanence properties for amenable inclusions. One is weak Haagerup property, the other is weak exactness.
{"title":"A note on relative amenability of finite von Neumann algebras","authors":"Xiaoyan Zhou, Junsheng Fang","doi":"10.7900/jot.2017dec06.2200","DOIUrl":"https://doi.org/10.7900/jot.2017dec06.2200","url":null,"abstract":"Let M be a finite von Neumann algebra (respectively, a type II1 factor) and let N⊂M be a II1 factor (respectively, N⊂M have an atomic part). We prove that if the inclusion N⊂M is amenable, then implies the identity map on M has an approximate factorization through Mm(C)⊗N via trace preserving normal unital completely positive maps, which is a generalization of a result of Haagerup. We also prove two permanence properties for amenable inclusions. One is weak Haagerup property, the other is weak exactness.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45714837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-15DOI: 10.7900/jot.2017nov27.2210
Jaewoong Kim, Jasang Yoon
n the first part of this paper, we introduce two notions of multivariable Duggal transforms (toral and spherical), and study their basic properties including hyponormality and norm-continuity. In the second part, we study how the Taylor spectrum and Taylor essential spectrum of 2-variable weighted shifts behave under the toral and spherical Duggal transforms including generalized Aluthge transforms. In the last part, we investigate nontrivial common invariant subspaces between the toral (respectively spherical) Duggal transform and the original n-tuple of bounded operators with dense ranges. We also study the sets of common invariant subspaces among them.
{"title":"Taylor spectra and common invariant subspaces through the Duggal and generalized Aluthge transforms for commuting n-tuples of operators","authors":"Jaewoong Kim, Jasang Yoon","doi":"10.7900/jot.2017nov27.2210","DOIUrl":"https://doi.org/10.7900/jot.2017nov27.2210","url":null,"abstract":"n the first part of this paper, we introduce two notions of multivariable Duggal transforms (toral and spherical), and study their basic properties including hyponormality and norm-continuity. In the second part, we study how the Taylor spectrum and Taylor essential spectrum of 2-variable weighted shifts behave under the toral and spherical Duggal transforms including generalized Aluthge transforms. In the last part, we investigate nontrivial common invariant subspaces between the toral (respectively spherical) Duggal transform and the original n-tuple of bounded operators with dense ranges. We also study the sets of common invariant subspaces among them.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44048837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-15DOI: 10.7900/jot.2017dec22.2179
Benoît R. Kloeckner
We propose a new approach to the spectral theory of perturbed linear operators in the case of a simple isolated eigenvalue. We obtain two kinds of results: ``radius bounds'' which ensure perturbation theory applies for perturbations up to an explicit size, and ``regularity bounds'' which control the variations of eigendata to any order. Our method is based on the implicit function theorem and proceeds by establishing differential inequalities on two natural quantities: the norm of the projection to the eigendirection, and the norm of the reduced resolvent. We obtain completely explicit results without any assumption on the underlying Banach space. In companion articles, on the one hand we apply the regularity bounds to Markov chains, obtaining non-asymptotic concentration and Berry-Esseen inequalities with explicit constants, and on the other hand we apply the radius bounds to transfer operators of intermittent maps, obtaining explicit high-temperature regimes where a spectral gap occurs.
{"title":"Effective perturbation theory for simple isolated eigenvalues of linear operators","authors":"Benoît R. Kloeckner","doi":"10.7900/jot.2017dec22.2179","DOIUrl":"https://doi.org/10.7900/jot.2017dec22.2179","url":null,"abstract":"We propose a new approach to the spectral theory of perturbed linear operators in the case of a simple isolated eigenvalue. We obtain two kinds of results: ``radius bounds'' which ensure perturbation theory applies for perturbations up to an explicit size, and ``regularity bounds'' which control the variations of eigendata to any order. Our method is based on the implicit function theorem and proceeds by establishing differential inequalities on two natural quantities: the norm of the projection to the eigendirection, and the norm of the reduced resolvent. We obtain completely explicit results without any assumption on the underlying Banach space. In companion articles, on the one hand we apply the regularity bounds to Markov chains, obtaining non-asymptotic concentration and Berry-Esseen inequalities with explicit constants, and on the other hand we apply the radius bounds to transfer operators of intermittent maps, obtaining explicit high-temperature regimes where a spectral gap occurs.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48253860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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