Pub Date : 2020-01-19DOI: 10.7900/jot.2020oct06.2319
N. Demni, Tarek Hamdi
In this paper, we derive explicit expressions for some classes of ⋆-moments of a free unitary Brownian motion compressed by a free projection, using various methods. While the moments of this nonnormal operator are readily derived through analytical or combinatorial methods, we only succeeded to derive its mixed ones after solving a nonlinear partial differential equation (pde) for their generating function. We shall also give some interest in odd alternating moments. In particular, we derive a linear pde for their generating function which we solve when the rank of the projection equals~1/2.
{"title":"On star moments of the compression of the free unitary Brownian motion by a free projection","authors":"N. Demni, Tarek Hamdi","doi":"10.7900/jot.2020oct06.2319","DOIUrl":"https://doi.org/10.7900/jot.2020oct06.2319","url":null,"abstract":"In this paper, we derive explicit expressions for some classes of ⋆-moments of a free unitary Brownian motion compressed by a free projection, using various methods. While the moments of this nonnormal operator are readily derived through analytical or combinatorial methods, we only succeeded to derive its mixed ones after solving a nonlinear partial differential equation (pde) for their generating function. We shall also give some interest in odd alternating moments. In particular, we derive a linear pde for their generating function which we solve when the rank of the projection equals~1/2.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42759970","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 : 2020-01-09DOI: 10.7900/jot.2019sep05.2324
Jurgen Muller, Maike Thelen
The Taylor (backward) shift on Bergman spaces Ap(Ω) for general open sets Ω in the extended complex plane shows rich variety concerning its dynamical behaviour. Different aspects are worked out, where in the case p<2 a recent result of Bayart and Matheron plays a central role.
{"title":"Dynamics of the Taylor shift on Bergman spaces","authors":"Jurgen Muller, Maike Thelen","doi":"10.7900/jot.2019sep05.2324","DOIUrl":"https://doi.org/10.7900/jot.2019sep05.2324","url":null,"abstract":"The Taylor (backward) shift on Bergman spaces Ap(Ω) for general open sets Ω in the extended complex plane shows rich variety concerning its dynamical behaviour. Different aspects are worked out, where in the case p<2 a recent result of Bayart and Matheron plays a central role.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45442065","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 : 2019-12-23DOI: 10.7900/jot.2020feb01.2285
Henning Bostelmann, D. Cadamuro, Gandalf Lechner
The wave operators W±(H1,H0) of two selfadjoint operators H0 and H1 are analyzed at asymptotic spectral values. Sufficient conditions for ∥(W±(H1,H0)−Pac1Pac0)f(H0)∥<∞ are given, where Pacj projects onto the subspace of absolutely continuous spectrum of Hj and f is an unbounded function (f-boundedness), both in the case of trace-class perturbations and in terms of the high-energy behaviour of the boundary values of the resolvent of H0 (smooth method). Examples include f-boundedness for the perturbed polyharmonic operator and for Schr"odinger operators with matrix-valued potentials. We discuss an application to the problem of quantum backflow.
{"title":"High energy bounds on wave operators","authors":"Henning Bostelmann, D. Cadamuro, Gandalf Lechner","doi":"10.7900/jot.2020feb01.2285","DOIUrl":"https://doi.org/10.7900/jot.2020feb01.2285","url":null,"abstract":"The wave operators W±(H1,H0) of two selfadjoint operators H0 and H1 are analyzed at asymptotic spectral values. Sufficient conditions for ∥(W±(H1,H0)−Pac1Pac0)f(H0)∥<∞ are given, where Pacj projects onto the subspace of absolutely continuous spectrum of Hj and f is an unbounded function (f-boundedness), both in the case of trace-class perturbations and in terms of the high-energy behaviour of the boundary values of the resolvent of H0 (smooth method). Examples include f-boundedness for the perturbed polyharmonic operator and for Schr\"odinger operators with matrix-valued potentials. We discuss an application to the problem of quantum backflow.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48187127","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 : 2019-12-15DOI: 10.7900/JOT.2018AUG31.2207
L. Molnár
We present characterizations of isomorphisms of Jordan algebras of quantum observables using only certain spectrum-preserving properties without assuming any kind of linearity. Similar problems involving the norm in the place of the spectrum are also studied.
{"title":"Spectral characterization of Jordan--Segal isomorphisms of quantum observables","authors":"L. Molnár","doi":"10.7900/JOT.2018AUG31.2207","DOIUrl":"https://doi.org/10.7900/JOT.2018AUG31.2207","url":null,"abstract":"We present characterizations of isomorphisms of Jordan algebras of quantum observables using only certain spectrum-preserving properties without assuming any kind of linearity. Similar problems involving the norm in the place of the spectrum are also studied.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48416987","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 : 2019-12-13DOI: 10.7900/jot.2019dec16.2278
Benjamin W. Passer, V. Paulsen
For finite-dimensional operator systems ST, T∈B(H)d, we show that the local lifting property and 1-exactness of ST may be characterized by measurements of the disparity between the matrix range W(T) and the minimal/maximal matrix convex sets over its individual levels. We then examine these concepts from the point of view of free spectrahedra, direct sums of operator systems, and products of matrix convex sets.
{"title":"Matrix range characterizations of operator system properties","authors":"Benjamin W. Passer, V. Paulsen","doi":"10.7900/jot.2019dec16.2278","DOIUrl":"https://doi.org/10.7900/jot.2019dec16.2278","url":null,"abstract":"For finite-dimensional operator systems ST, T∈B(H)d, we show that the local lifting property and 1-exactness of ST may be characterized by measurements of the disparity between the matrix range W(T) and the minimal/maximal matrix convex sets over its individual levels. We then examine these concepts from the point of view of free spectrahedra, direct sums of operator systems, and products of matrix convex sets.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46880367","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 : 2019-12-12DOI: 10.7900/jot.2019dec20.2261
Brianna Leary
We investigate the use of Popa's asymptotic orthogonality to establish maximal amenability for amalgamated free product von Neumann algebras. Although new techniques have recently been developed to consider amalgamated free products, we find that the technique developed by Popa in the 1980s can be used to demonstrate maximal amenability for a certain family of amalgamated free product von Neumann algebras.
{"title":"Maximal amenability with asymptotic orthogonality in amalgamated free products","authors":"Brianna Leary","doi":"10.7900/jot.2019dec20.2261","DOIUrl":"https://doi.org/10.7900/jot.2019dec20.2261","url":null,"abstract":"We investigate the use of Popa's asymptotic orthogonality to establish maximal amenability for amalgamated free product von Neumann algebras. Although new techniques have recently been developed to consider amalgamated free products, we find that the technique developed by Popa in the 1980s can be used to demonstrate maximal amenability for a certain family of amalgamated free product von Neumann algebras.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45557826","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 : 2019-12-10DOI: 10.7900/jot.2020feb06.2279
Keshab Chandra Bakshi, Vijay Kodiyalam
We show a close relationship between non-degenerate smooth commuting squares of II1-factors with horizontal inclusions of finite index and inclusions of subfactor planar algebras by showing that each leads to a construction of the other. One direction of this uses the Guionnet--Jones--Shlyakhtenko construction.
{"title":"Commuting squares and planar subalgebras","authors":"Keshab Chandra Bakshi, Vijay Kodiyalam","doi":"10.7900/jot.2020feb06.2279","DOIUrl":"https://doi.org/10.7900/jot.2020feb06.2279","url":null,"abstract":"We show a close relationship between non-degenerate smooth commuting squares of II1-factors with horizontal inclusions of finite index and inclusions of subfactor planar algebras by showing that each leads to a construction of the other. One direction of this uses the Guionnet--Jones--Shlyakhtenko construction.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46255805","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 : 2019-12-08DOI: 10.7900/jot.2019nov12.2262
A. Mal, K. Paul
This paper deals with the study of Birkhoff--James orthogonality of a linear operator to a subspace of operators defined between arbitrary Banach spaces. In case the domain space is reflexive and the subspace is finite dimensional we obtain a complete characterization. For arbitrary Banach spaces, we obtain the same under some additional conditions. For an arbitrary Hilbert space H, we also study orthogonality to a subspace of the space of linear operators L(H), both with respect to operator norm as well as numerical radius norm.
{"title":"Birkhoff--James orthogonality to a subspace of operators defined between Banach spaces","authors":"A. Mal, K. Paul","doi":"10.7900/jot.2019nov12.2262","DOIUrl":"https://doi.org/10.7900/jot.2019nov12.2262","url":null,"abstract":"This paper deals with the study of Birkhoff--James orthogonality of a linear operator to a subspace of operators defined between arbitrary Banach spaces. In case the domain space is reflexive and the subspace is finite dimensional we obtain a complete characterization. For arbitrary Banach spaces, we obtain the same under some additional conditions. For an arbitrary Hilbert space H, we also study orthogonality to a subspace of the space of linear operators L(H), both with respect to operator norm as well as numerical radius norm.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41653115","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 : 2019-11-27DOI: 10.7900/jot.2019dec04.2277
Malte Gerhold, O. Shalit
This note treats a simple minded question: textit{what does a typical random matrix range look like?} We study the relationship between various modes of convergence for tuples of operators on the one hand, and continuity of matrix ranges with respect to the Hausdorff metric on the other. In particular, we show that the matrix range of a tuple generating a continuous field of C∗-algebras is continuous in the sense that every level is continuous in the Hausdorff metric. Using this observation together with known results on strong convergence in distribution of matrix ensembles, we identify the limit matrix ranges to which the matrix ranges of independent Wigner or Haar ensembles converge.
{"title":"On the matrix range of random matrices","authors":"Malte Gerhold, O. Shalit","doi":"10.7900/jot.2019dec04.2277","DOIUrl":"https://doi.org/10.7900/jot.2019dec04.2277","url":null,"abstract":"This note treats a simple minded question: textit{what does a typical random matrix range look like?} We study the relationship between various modes of convergence for tuples of operators on the one hand, and continuity of matrix ranges with respect to the Hausdorff metric on the other. In particular, we show that the matrix range of a tuple generating a continuous field of C∗-algebras is continuous in the sense that every level is continuous in the Hausdorff metric. Using this observation together with known results on strong convergence in distribution of matrix ensembles, we identify the limit matrix ranges to which the matrix ranges of independent Wigner or Haar ensembles converge.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45985213","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 : 2019-11-15DOI: 10.7900/jot.2020aug03.2301
Joao R. Carmo, S. Noor
A Hilbert space operator U is called textit{universal} (in the sense of Rota) if every Hilbert space operator is similar to a multiple of U restricted to one of its invariant subspaces. It follows that the textit{invariant subspace problem} for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for U are one dimensional. In this article we characterize all linear fractional composition operators Cϕf=f∘ϕ that have universal translates on both the classical Hardy spaces H2(C+) and H2(D) of the half-plane and the unit disk, respectively. The new example here is the composition operator on H2(D) with affine symbol ϕa(z)=az+(1−a) for $0
{"title":"Universal composition operators","authors":"Joao R. Carmo, S. Noor","doi":"10.7900/jot.2020aug03.2301","DOIUrl":"https://doi.org/10.7900/jot.2020aug03.2301","url":null,"abstract":"A Hilbert space operator U is called textit{universal} (in the sense of Rota) if every Hilbert space operator is similar to a multiple of U restricted to one of its invariant subspaces. It follows that the textit{invariant subspace problem} for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for U are one dimensional. In this article we characterize all linear fractional composition operators Cϕf=f∘ϕ that have universal translates on both the classical Hardy spaces H2(C+) and H2(D) of the half-plane and the unit disk, respectively. The new example here is the composition operator on H2(D) with affine symbol ϕa(z)=az+(1−a) for $0","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44917852","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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