Pub Date : 2021-11-15DOI: 10.7900/JOT.2020JUN08.2304
M. L. Cristoforis, P. Musolino, J. Taskinen
As is well known, by the Floquet--Bloch theory for periodic problems, one can transform a spectral Laplace--Dirichlet problem in the plane with a set of periodic perforations into a family of ``model problems'' depending on a parameter η∈[0,2π]2 for quasiperiodic functions in the unit cell with a single perforation. We prove real analyticity results for the eigenvalues of the model problems upon perturbation of the shape of the perforation of the unit~cell.
{"title":"A real analyticity result for symmetric functions of the eigenvalues of a quasiperiodic spectral problem for the Dirichlet Laplacian","authors":"M. L. Cristoforis, P. Musolino, J. Taskinen","doi":"10.7900/JOT.2020JUN08.2304","DOIUrl":"https://doi.org/10.7900/JOT.2020JUN08.2304","url":null,"abstract":"As is well known, by the Floquet--Bloch theory for periodic problems, one can transform a spectral Laplace--Dirichlet problem in the plane with a set of periodic perforations into a family of ``model problems'' depending on a parameter η∈[0,2π]2 for quasiperiodic functions in the unit cell with a single perforation. We prove real analyticity results for the eigenvalues of the model problems upon perturbation of the shape of the perforation of the unit~cell.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46627489","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 : 2021-11-15DOI: 10.7900/jot.2020apr11.2270
F. Botelho, D. Ilišević
In this paper we investigate inverse eigenvalue problems for finite spectrum linear isometries on complex Banach spaces. We establish necessary conditions on a finite set of modulus one complex numbers to be the spectrum of a linear isometry. In particular, we study periodic linear isometries on the large class of Banach spaces X with the following property: if T:X→X is a linear isometry with two-point spectrum {1,λ} then λ=−1 or the eigenprojections of T are Hermitian.
{"title":"On isometries with finite spectrum","authors":"F. Botelho, D. Ilišević","doi":"10.7900/jot.2020apr11.2270","DOIUrl":"https://doi.org/10.7900/jot.2020apr11.2270","url":null,"abstract":"In this paper we investigate inverse eigenvalue problems for finite spectrum linear isometries on complex Banach spaces. We establish necessary conditions on a finite set of modulus one complex numbers to be the spectrum of a linear isometry. In particular, we study periodic linear isometries on the large class of Banach spaces X with the following property: if T:X→X is a linear isometry with two-point spectrum {1,λ} then λ=−1 or the eigenprojections of T are Hermitian.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42029020","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 : 2021-06-15DOI: 10.7900/jot.2020jan22.2275
M. Gerasimova, A. Thom
In this note we consider a p-isometrisability property of discrete groups. If p=2 this property is equivalent to the well-studied notion of unitarisability. We prove that amenable groups are p-isometrisable for all p∈(1,∞). Conversely, we show that every group containing a non-abelian free subgroup is not p-isometrisable for any p∈(1,∞). We also discuss some open questions and possible relations of p-isometrisability with the recently introduced Littlewood exponent Lit(Γ).
{"title":"On the isometrisability of group representations on p-spaces","authors":"M. Gerasimova, A. Thom","doi":"10.7900/jot.2020jan22.2275","DOIUrl":"https://doi.org/10.7900/jot.2020jan22.2275","url":null,"abstract":"In this note we consider a p-isometrisability property of discrete groups. If p=2 this property is equivalent to the well-studied notion of unitarisability. We prove that amenable groups are p-isometrisable for all p∈(1,∞). Conversely, we show that every group containing a non-abelian free subgroup is not p-isometrisable for any p∈(1,∞). We also discuss some open questions and possible relations of p-isometrisability with the recently introduced Littlewood exponent Lit(Γ).","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42343999","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 : 2021-06-15DOI: 10.7900/jot.2020mar09.2282
Yongle Jiang
We prove that L(SL2(k)) is a maximal Haagerup--von Neumann subalgebra in L(k2⋊SL2(k)) for k=Q and k=Z. The key step for the proof is a complete description of all intermediate von Neumann subalgebras between L(SL2(k)) and L∞(Y)⋊SL2(k), where SL2(k)↷Y denotes the quotient of the algebraic action SL2(k)↷ˆk2 by modding out the relation ϕ∼ϕ′, where ϕ, ϕ′∈ˆk2 and ϕ′(x,y):=ϕ(−x,−y) for all (x,y)∈k2. As a by-product, we show L(PSL2(Q)) is a maximal von Neumann subalgebra in L∞(Y)⋊PSL2(Q); in particular, PSL2(Q)↷Y is a prime action.
{"title":"Maximal Haagerup subalgebras in L(Z2⋊SL2(Z))","authors":"Yongle Jiang","doi":"10.7900/jot.2020mar09.2282","DOIUrl":"https://doi.org/10.7900/jot.2020mar09.2282","url":null,"abstract":"We prove that L(SL2(k)) is a maximal Haagerup--von Neumann subalgebra in L(k2⋊SL2(k)) for k=Q and k=Z. The key step for the proof is a complete description of all intermediate von Neumann subalgebras between L(SL2(k)) and L∞(Y)⋊SL2(k), where SL2(k)↷Y denotes the quotient of the algebraic action SL2(k)↷ˆk2 by modding out the relation ϕ∼ϕ′, where ϕ, ϕ′∈ˆk2 and ϕ′(x,y):=ϕ(−x,−y) for all (x,y)∈k2. As a by-product, we show L(PSL2(Q)) is a maximal von Neumann subalgebra in L∞(Y)⋊PSL2(Q); in particular, PSL2(Q)↷Y is a prime action.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71360555","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 : 2021-06-15DOI: 10.7900/jot.2020mar23.2288
F. Bayart, Maofa Wang, Xingxing Yao
On the Hilbert space of Dirichlet series with square summable coefficients, we show that the linear combinations of two composition operators induced by linear symbols are compact only when each one of them is compact. Moreover, such rigid behavior holds partially for some more general symbols.
{"title":"Linear combinations of composition operators with linear symbols on a Hilbert space of Dirichlet series","authors":"F. Bayart, Maofa Wang, Xingxing Yao","doi":"10.7900/jot.2020mar23.2288","DOIUrl":"https://doi.org/10.7900/jot.2020mar23.2288","url":null,"abstract":"On the Hilbert space of Dirichlet series with square summable coefficients, we show that the linear combinations of two composition operators induced by linear symbols are compact only when each one of them is compact. Moreover, such rigid behavior holds partially for some more general symbols.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71360561","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 : 2021-03-15DOI: 10.7900/jot.2019nov12.2291
Iakovos Androulidakis, Omar Mohsen, Robert Yuncken
Motivated by the study of H"ormander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized functions on the underlying manifold, and on the leaves and their holonomy covers. This generalizes Schwartz kernel operators to singular foliations. We also define the algebra of smoothing operators in this context and prove that it is a two-sided ideal.
{"title":"The convolution algebra of Schwartz kernels along a singular foliation","authors":"Iakovos Androulidakis, Omar Mohsen, Robert Yuncken","doi":"10.7900/jot.2019nov12.2291","DOIUrl":"https://doi.org/10.7900/jot.2019nov12.2291","url":null,"abstract":"Motivated by the study of H\"ormander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized functions on the underlying manifold, and on the leaves and their holonomy covers. This generalizes Schwartz kernel operators to singular foliations. We also define the algebra of smoothing operators in this context and prove that it is a two-sided ideal.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46666098","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 : 2021-03-15DOI: 10.7900/jot.2019jul22.2284
Kit C. Chan, Rebecca Sanders
Taking the perspective that a bilateral weighted shift is an operator that shifts some two-sided canonical basic sequence of ℓp(Z), with 1⩽p<∞, we show that every bilateral weighted shift on ℓp(Z) has a factorization T=AB, where A and B are hypercyclic bilateral weighted shifts. For the case when T is invertible, both shifts A and B may be selected to be invertible as well. Moreover, we show analogous hypercyclic factorization results for diagonal operators with nonzero diagonal entries.
{"title":"Hypercyclic shift factorizations for bilateral weighted shift operators","authors":"Kit C. Chan, Rebecca Sanders","doi":"10.7900/jot.2019jul22.2284","DOIUrl":"https://doi.org/10.7900/jot.2019jul22.2284","url":null,"abstract":"Taking the perspective that a bilateral weighted shift is an operator that shifts some two-sided canonical basic sequence of ℓp(Z), with 1⩽p<∞, we show that every bilateral weighted shift on ℓp(Z) has a factorization T=AB, where A and B are hypercyclic bilateral weighted shifts. For the case when T is invertible, both shifts A and B may be selected to be invertible as well. Moreover, we show analogous hypercyclic factorization results for diagonal operators with nonzero diagonal entries.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49343105","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 : 2021-03-15DOI: 10.7900/jot.2019sep21.2256
E. Gallardo-Gutiérrez, Miguel Monsalve-López
It is proved that a wide class of Bishop-type operators Tϕ,τ are power-regular operators in Lp(Ω,μ), 1⩽p<∞, computing the exact value of the local spectral radius at any function u∈Lp(Ω,μ). Moreover, it is shown that the local spectral radius at any u coincides with the spectral radius of Tϕ,τ as far as u is non-zero. As a consequence, it is proved that non-invertible Bishop-type operators are non-decomposable whenever log|ϕ|∈L1(Ω,μ) (in particular, not quasinilpotent); not enjoying even the weaker spectral decompositions textit{Bishop property} (β) and textit{property} (δ).
{"title":"Power-regular Bishop operators and spectral decompositions","authors":"E. Gallardo-Gutiérrez, Miguel Monsalve-López","doi":"10.7900/jot.2019sep21.2256","DOIUrl":"https://doi.org/10.7900/jot.2019sep21.2256","url":null,"abstract":"It is proved that a wide class of Bishop-type operators Tϕ,τ are power-regular operators in Lp(Ω,μ), 1⩽p<∞, computing the exact value of the local spectral radius at any function u∈Lp(Ω,μ). Moreover, it is shown that the local spectral radius at any u coincides with the spectral radius of Tϕ,τ as far as u is non-zero. As a consequence, it is proved that non-invertible Bishop-type operators are non-decomposable whenever log|ϕ|∈L1(Ω,μ) (in particular, not quasinilpotent); not enjoying even the weaker spectral decompositions textit{Bishop property} (β) and textit{property} (δ).","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44786293","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 : 2021-01-27DOI: 10.7900/jot.2021jan26.2316
Sahiba Arora
We initiate a theory of locally eventually positive operator semigroups on Banach lattices. Intuitively this means: given a positive initial datum, the solution of the corresponding Cauchy problem becomes (and stays) positive in a part of the domain, after a sufficiently large time. A drawback of the present theory of eventually positive C0-semigroups is that it is applicable only when the leading eigenvalue of the semigroup generator has a strongly positive eigenvector. We weaken this requirement and give sufficient criteria for individual and uniform local eventual positivity of the semigroup. This allows us to treat a larger class of examples by giving us more freedom on the domain when dealing with function spaces − for instance, the square of the Laplace operator with Dirichlet boundary conditions on L2 and the Dirichlet bi-Laplacian on Lp-spaces. Besides, we establish various spectral and convergence properties of locally eventually positive semigroups.
{"title":"Locally eventually positive operator semigroups","authors":"Sahiba Arora","doi":"10.7900/jot.2021jan26.2316","DOIUrl":"https://doi.org/10.7900/jot.2021jan26.2316","url":null,"abstract":"We initiate a theory of locally eventually positive operator semigroups on Banach lattices. Intuitively this means: given a positive initial datum, the solution of the corresponding Cauchy problem becomes (and stays) positive in a part of the domain, after a sufficiently large time. A drawback of the present theory of eventually positive C0-semigroups is that it is applicable only when the leading eigenvalue of the semigroup generator has a strongly positive eigenvector. We weaken this requirement and give sufficient criteria for individual and uniform local eventual positivity of the semigroup. This allows us to treat a larger class of examples by giving us more freedom on the domain when dealing with function spaces − for instance, the square of the Laplace operator with Dirichlet boundary conditions on L2 and the Dirichlet bi-Laplacian on Lp-spaces. Besides, we establish various spectral and convergence properties of locally eventually positive semigroups.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47046795","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 : 2021-01-20DOI: 10.7900/jot.2021jan21.2309
R. Chill, Sebastian Król
We characterise the Kato property of a sectorial form a, defined on a Hilbert space V, with respect to a larger Hilbert space H in terms of two bounded, selfadjoint operators T and Q determined by the imaginary part of a and the embedding of V into H, respectively. As a consequence, we show that if a bounded selfadjoint operator T on a Hilbert space V is in the Schatten class Sp(V) (p⩾1), then the associated form aT(⋅,⋅):=⟨(I+iT)⋅,⋅⟩V has the Kato property with respect to every Hilbert space H into which V is densely and continuously embedded. This result is in a sense sharp. Another result says that if T and Q commute then the form a with respect to H possesses the Kato property.
{"title":"Note on the Kato property of sectorial forms","authors":"R. Chill, Sebastian Król","doi":"10.7900/jot.2021jan21.2309","DOIUrl":"https://doi.org/10.7900/jot.2021jan21.2309","url":null,"abstract":"We characterise the Kato property of a sectorial form a, defined on a Hilbert space V, with respect to a larger Hilbert space H in terms of two bounded, selfadjoint operators T and Q determined by the imaginary part of a and the embedding of V into H, respectively. As a consequence, we show that if a bounded selfadjoint operator T on a Hilbert space V is in the Schatten class Sp(V) (p⩾1), then the associated form aT(⋅,⋅):=⟨(I+iT)⋅,⋅⟩V has the Kato property with respect to every Hilbert space H into which V is densely and continuously embedded. This result is in a sense sharp. Another result says that if T and Q commute then the form a with respect to H possesses the Kato property.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41562850","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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