Abstract Matrix-valued asymmetric truncated Toeplitz operators are compressions of multiplication operators acting between two model spaces. These are the generalization of matrix-valued truncated Toeplitz operators. In this article, we describe symbols of matrix-valued asymmetric truncated Toeplitz operators equal to the zero operator. We also use generalized Crofoot transform to find a connection between the symbols of matrix-valued asymmetric truncated Toeplitz operators T ( Θ 1 , Θ 2 ) {mathcal{T}}left({Theta }_{1},{Theta }_{2}) and T ( Θ 1 ′ , Θ 2 ′ ) {mathcal{T}}left({Theta }_{1}^{^{prime} },{Theta }_{2}^{^{prime} }) .
{"title":"Generalized Crofoot transform and applications","authors":"Rewayat Khan, A. Farooq","doi":"10.1515/conop-2022-0138","DOIUrl":"https://doi.org/10.1515/conop-2022-0138","url":null,"abstract":"Abstract Matrix-valued asymmetric truncated Toeplitz operators are compressions of multiplication operators acting between two model spaces. These are the generalization of matrix-valued truncated Toeplitz operators. In this article, we describe symbols of matrix-valued asymmetric truncated Toeplitz operators equal to the zero operator. We also use generalized Crofoot transform to find a connection between the symbols of matrix-valued asymmetric truncated Toeplitz operators T ( Θ 1 , Θ 2 ) {mathcal{T}}left({Theta }_{1},{Theta }_{2}) and T ( Θ 1 ′ , Θ 2 ′ ) {mathcal{T}}left({Theta }_{1}^{^{prime} },{Theta }_{2}^{^{prime} }) .","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45662702","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}
Alexander Caicedo, J. Ramos-Fernández, M. Salas-Brown
Abstract In this article, all sequences u {boldsymbol{u}} , v {boldsymbol{v}} , and w {boldsymbol{w}} that define continuous and compact tridiagonal operators T u , v , w {T}_{u,v,w} acting on the weighted sequence space l β 2 {l}_{beta }^{2} were characterized. Additionally, the essential norm of this operator, and as an important consequence of our results, the essential norm of multiplication operator M u {M}_{u} acting on l β 2 {l}_{beta }^{2} spaces was calculated.
{"title":"On the compactness and the essential norm of operators defined by infinite tridiagonal matrices","authors":"Alexander Caicedo, J. Ramos-Fernández, M. Salas-Brown","doi":"10.1515/conop-2022-0143","DOIUrl":"https://doi.org/10.1515/conop-2022-0143","url":null,"abstract":"Abstract In this article, all sequences u {boldsymbol{u}} , v {boldsymbol{v}} , and w {boldsymbol{w}} that define continuous and compact tridiagonal operators T u , v , w {T}_{u,v,w} acting on the weighted sequence space l β 2 {l}_{beta }^{2} were characterized. Additionally, the essential norm of this operator, and as an important consequence of our results, the essential norm of multiplication operator M u {M}_{u} acting on l β 2 {l}_{beta }^{2} spaces was calculated.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"10 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41670534","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 article, we considered the generalized Hausdorff operator ℋμ,ϕ,a {{mathcal{ {mathcal H} }}}_{mu ,phi ,a} on Bergmann space and determined the conditions on ϕ phi and a a so that the operator is bounded. In addition, we studied the action of the Hausdorff operator on the truncated domain to estimate norm ℋμ,ϕ,a {{mathcal{ {mathcal H} }}}_{mu ,phi ,a} and established a relation with quasi-Hausdorff operator on Bergmann space.
{"title":"Generalized Hausdorff operator on Bergmann spaces","authors":"Sasikala Perumal, Kalaivani Kamalakkannan","doi":"10.1515/conop-2023-0101","DOIUrl":"https://doi.org/10.1515/conop-2023-0101","url":null,"abstract":"Abstract In this article, we considered the generalized Hausdorff operator <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">ℋ</m:mi> </m:mrow> <m:mrow> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>ϕ</m:mi> <m:mo>,</m:mo> <m:mi>a</m:mi> </m:mrow> </m:msub> </m:math> {{mathcal{ {mathcal H} }}}_{mu ,phi ,a} on Bergmann space and determined the conditions on <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ϕ</m:mi> </m:math> phi and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>a</m:mi> </m:math> a so that the operator is bounded. In addition, we studied the action of the Hausdorff operator on the truncated domain to estimate norm <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">ℋ</m:mi> </m:mrow> <m:mrow> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>ϕ</m:mi> <m:mo>,</m:mo> <m:mi>a</m:mi> </m:mrow> </m:msub> </m:math> {{mathcal{ {mathcal H} }}}_{mu ,phi ,a} and established a relation with quasi-Hausdorff operator on Bergmann space.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135705256","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 article, we define a new subclass of analytic functions ℛ ( t , δ ) {mathcal{ {mathcal R} }}left(t,delta ) in the open unit disk D {mathbb{D}} associated with the petal-shaped domain. The bounds of the coefficients a 2 {a}_{2} , a 3 {a}_{3} , and a 4 {a}_{4} for the functions in the new class are obtained. We also acquire the bound of the Fekete-Szegö inequality and the bound of the Toeplitz determinants T 2 ( 2 ) {{mathcal{T}}}_{2}left(2) and T 3 ( 1 ) {{mathcal{T}}}_{3}left(1) for the functions in the defined class.
{"title":"Estimation of coefficient bounds for a subclass of Sakaguchi kind functions mapped onto various domains","authors":"B. Aarthy, B. Keerthi","doi":"10.1515/conop-2022-0140","DOIUrl":"https://doi.org/10.1515/conop-2022-0140","url":null,"abstract":"Abstract In this article, we define a new subclass of analytic functions ℛ ( t , δ ) {mathcal{ {mathcal R} }}left(t,delta ) in the open unit disk D {mathbb{D}} associated with the petal-shaped domain. The bounds of the coefficients a 2 {a}_{2} , a 3 {a}_{3} , and a 4 {a}_{4} for the functions in the new class are obtained. We also acquire the bound of the Fekete-Szegö inequality and the bound of the Toeplitz determinants T 2 ( 2 ) {{mathcal{T}}}_{2}left(2) and T 3 ( 1 ) {{mathcal{T}}}_{3}left(1) for the functions in the defined class.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45305020","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 Given Banach space operators S i {S}_{i} and T i {T}_{i} , i = 1 , 2 i=1,2 , we use elementary properties of the left and right multiplication operators to prove, that if the tensor products pair ( S 1 ⊗ S 2 , T 1 ⊗ T 2 ) left({S}_{1}otimes {S}_{2},{T}_{1}otimes {T}_{2}) is strictly m m -isometric, i.e., Δ S 1 ⊗ S 2 , T 1 ⊗ T 2 m ( I ⊗ I ) = ∑ j = 0 m ( − 1 ) j m j ( S 1 ⊗ S 2 ) m − j ( T 1 ⊗ T 2 ) m − j = 0 {Delta }_{{S}_{1}otimes {S}_{2},{T}_{1}otimes {T}_{2}}^{m}left(Iotimes I)={sum }_{j=0}^{m}{left(-1)}^{j}left(begin{array}{c}m jend{array}right){left({S}_{1}otimes {S}_{2})}^{m-j}{left({T}_{1}otimes {T}_{2})}^{m-j}=0 , then there exist a non-zero scalar c c and positive integers m 1 , m 2 ≤ m {m}_{1},{m}_{2}le m such that m = m 1 + m 2 − 1 m={m}_{1}+{m}_{2}-1 , ( S 1 , c T 1 ) left({S}_{1},c{T}_{1}) is strict- m 1 {m}_{1} -isometric and S 2 , 1 c T 2 left({S}_{2},frac{1}{c}{T}_{2}right) is strict m 2 {m}_{2} -isometric.
{"title":"m-Isometric tensor products","authors":"Bhagawati Prashad Duggal, I. Kim","doi":"10.1515/conop-2022-0142","DOIUrl":"https://doi.org/10.1515/conop-2022-0142","url":null,"abstract":"Abstract Given Banach space operators S i {S}_{i} and T i {T}_{i} , i = 1 , 2 i=1,2 , we use elementary properties of the left and right multiplication operators to prove, that if the tensor products pair ( S 1 ⊗ S 2 , T 1 ⊗ T 2 ) left({S}_{1}otimes {S}_{2},{T}_{1}otimes {T}_{2}) is strictly m m -isometric, i.e., Δ S 1 ⊗ S 2 , T 1 ⊗ T 2 m ( I ⊗ I ) = ∑ j = 0 m ( − 1 ) j m j ( S 1 ⊗ S 2 ) m − j ( T 1 ⊗ T 2 ) m − j = 0 {Delta }_{{S}_{1}otimes {S}_{2},{T}_{1}otimes {T}_{2}}^{m}left(Iotimes I)={sum }_{j=0}^{m}{left(-1)}^{j}left(begin{array}{c}m jend{array}right){left({S}_{1}otimes {S}_{2})}^{m-j}{left({T}_{1}otimes {T}_{2})}^{m-j}=0 , then there exist a non-zero scalar c c and positive integers m 1 , m 2 ≤ m {m}_{1},{m}_{2}le m such that m = m 1 + m 2 − 1 m={m}_{1}+{m}_{2}-1 , ( S 1 , c T 1 ) left({S}_{1},c{T}_{1}) is strict- m 1 {m}_{1} -isometric and S 2 , 1 c T 2 left({S}_{2},frac{1}{c}{T}_{2}right) is strict m 2 {m}_{2} -isometric.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41973409","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 A one-variable Hankel matrix H a {H}_{a} is an infinite matrix H a = [ a ( i + j ) ] i , j ≥ 0 {H}_{a}={left[aleft(i+j)]}_{i,jge 0} . Similarly, for any d ≥ 2 dge 2 , a d d -variable Hankel matrix is defined as H a = [ a ( i + j ) ] {H}_{{bf{a}}}=left[{bf{a}}left({bf{i}}+{bf{j}})] , where i = ( i 1 , … , i d ) {bf{i}}=left({i}_{1},ldots ,{i}_{d}) and j = ( j 1 , … , j d ) {bf{j}}=left({j}_{1},ldots ,{j}_{d}) , with i 1 , … , i d , j 1 , … , j d ≥ 0 {i}_{1},ldots ,{i}_{d},{j}_{1},ldots ,{j}_{d}ge 0 . For γ > 0 gamma gt 0 , Pushnitski and Yafaev proved that the eigenvalues of the compact one-variable Hankel matrices H a {H}_{a} with a ( j ) = j − 1 ( log j ) − γ aleft(j)={j}^{-1}{left(log j)}^{-gamma } , for j ≥ 2 jge 2 , obey the asymptotics λ n ( H a ) ∼ C γ n − γ {lambda }_{n}left({H}_{a})hspace{0.33em} sim hspace{0.33em}{C}_{gamma }{n}^{-gamma } , as n → + ∞ nto +infty , where the constant C γ {C}_{gamma } is calculated explicitly. This article presents the following d d -variable analogue. Let γ > 0 gamma gt 0 and a ( j ) = j − d ( log j ) − γ aleft(j)={j}^{-d}{left(log j)}^{-gamma } , for j ≥ 2 jge 2 . If a ( j 1 , … , j d ) = a ( j 1 + ⋯ + j d ) {bf{a}}left({j}_{1},ldots ,{j}_{d})=aleft({j}_{1}+cdots +{j}_{d}) , then H a {H}_{{bf{a}}} is compact and its eigenvalues follow the asymptotics λ n ( H a ) ∼ C d , γ n − γ {lambda }_{n}left({H}_{{bf{a}}})hspace{0.33em} sim hspace{0.33em}{C}_{d,gamma }{n}^{-gamma } , as n → + ∞ nto +infty , where the constant C d , γ {C}_{d,gamma } is calculated explicitly.
摘要单变量汉克尔矩阵H A {H＿a}是一个无限矩阵H A = [A (i + j)] i,j≥0 {H＿a}= {}{}{left[aleft(i+j)]} ＿i,j{ge 0}。同样，对于任意d≥2 dge 2, d变量汉克尔矩阵定义为Ha = [a (i + j)] {H＿}={{bf{a}}}left[{bf{a}}left({bf{i}}+{bf{j}})]，其中i = (i 1，…，i d) {bf{i}}=left ({i＿1}, {}ldots,{i＿d})，j = (j 1，…，j d) {=}{bf{j}}left ({j＿1}, {}ldots,{j＿d})，其中i 1，…，i d,j 1，…，j d≥0 {i＿1}, {}{}ldots,{i＿d},{j＿1}, {}{}ldots,{j＿d}{}ge 0。对于γ > gammagt 0, Pushnitski和Yafaev证明了a (j)=j−1 (log j)−γ a {}{}left (j)=j^-1 {}{}{left (log j}){^-gamma，}对于j≥2 j ge 2，服从渐近性λ n (H a) ~ C γ n−γ {lambda ＿n}{}left (H＿a){}{}hspace{0.33em}sim ＿hspace{0.33em}{C}{gamma n}{^}-{gamma，}为n→+∞nto + infty，其中常数C γ {C＿}{gamma显式计算。本文介绍了下面的d变量模拟。设γ > 0}gammagt 0, a (j)=j−d (log j)−γ a left (j)={j}^{-d}{left (log j)}^{- gamma，}对于j≥2 jge 2。如果a (j 1，…，j d)=a (j 1+⋯+j d) {bf{a}}left ({j＿1}, {}ldots,{j＿d})=a {}left ({j＿1}+ {}cdots +{j＿d})，则H a {H＿}是紧致的，其特征值遵循渐近性λ n (H a) ～ C d， γ n−{γ }{{bf{a}}}{lambda ＿n}{}left (H＿{)}{{bf{a}}}hspace{0.33em}sim ＿dhspace{0.33em}{C}, {gamma n}{^}-{gamma，为}n→+∞nto + infty，其中常数C d， γ {C＿d}, {gamma是显式}计算的。
{"title":"Eigenvalue asymptotics for a class of multi-variable Hankel matrices","authors":"Christos Panagiotis Tantalakis","doi":"10.1515/conop-2022-0137","DOIUrl":"https://doi.org/10.1515/conop-2022-0137","url":null,"abstract":"Abstract A one-variable Hankel matrix H a {H}_{a} is an infinite matrix H a = [ a ( i + j ) ] i , j ≥ 0 {H}_{a}={left[aleft(i+j)]}_{i,jge 0} . Similarly, for any d ≥ 2 dge 2 , a d d -variable Hankel matrix is defined as H a = [ a ( i + j ) ] {H}_{{bf{a}}}=left[{bf{a}}left({bf{i}}+{bf{j}})] , where i = ( i 1 , … , i d ) {bf{i}}=left({i}_{1},ldots ,{i}_{d}) and j = ( j 1 , … , j d ) {bf{j}}=left({j}_{1},ldots ,{j}_{d}) , with i 1 , … , i d , j 1 , … , j d ≥ 0 {i}_{1},ldots ,{i}_{d},{j}_{1},ldots ,{j}_{d}ge 0 . For γ > 0 gamma gt 0 , Pushnitski and Yafaev proved that the eigenvalues of the compact one-variable Hankel matrices H a {H}_{a} with a ( j ) = j − 1 ( log j ) − γ aleft(j)={j}^{-1}{left(log j)}^{-gamma } , for j ≥ 2 jge 2 , obey the asymptotics λ n ( H a ) ∼ C γ n − γ {lambda }_{n}left({H}_{a})hspace{0.33em} sim hspace{0.33em}{C}_{gamma }{n}^{-gamma } , as n → + ∞ nto +infty , where the constant C γ {C}_{gamma } is calculated explicitly. This article presents the following d d -variable analogue. Let γ > 0 gamma gt 0 and a ( j ) = j − d ( log j ) − γ aleft(j)={j}^{-d}{left(log j)}^{-gamma } , for j ≥ 2 jge 2 . If a ( j 1 , … , j d ) = a ( j 1 + ⋯ + j d ) {bf{a}}left({j}_{1},ldots ,{j}_{d})=aleft({j}_{1}+cdots +{j}_{d}) , then H a {H}_{{bf{a}}} is compact and its eigenvalues follow the asymptotics λ n ( H a ) ∼ C d , γ n − γ {lambda }_{n}left({H}_{{bf{a}}})hspace{0.33em} sim hspace{0.33em}{C}_{d,gamma }{n}^{-gamma } , as n → + ∞ nto +infty , where the constant C d , γ {C}_{d,gamma } is calculated explicitly.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45005453","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 Let E E be a complex Banach lattice and T T is an operator in the center Z ( E ) = { T : ∣ T ∣ ≤ λ I for some λ } Zleft(E)=left{T:| T| le lambda Ihspace{0.33em}hspace{0.1em}text{for some}hspace{0.1em}hspace{0.33em}lambda right} of E E . Then, the essential norm ‖ T ‖ e Vert T{Vert }_{e} of T T equals the essential spectral radius r e ( T ) {r}_{e}left(T) of T T . We also prove r e ( T ) = max { ‖ T A d ‖ , r e ( T A ) } {r}_{e}left(T)=max left{Vert {T}_{}hspace{-0.35em}{}_{{A}^{d}}Vert ,{r}_{e}left({T}_{A})right} , where T A {T}_{A} is the atomic part of T T and T A d {T}_{}hspace{-0.35em}{}_{{A}^{d}} is the nonatomic part of T T . Moreover, r e ( T A ) = limsup ℱ λ a {r}_{e}left({T}_{A})={mathrm{limsup}}_{{mathcal{ {mathcal F} }}}{lambda }_{a} , where ℱ {mathcal{ {mathcal F} }} is the Fréchet filter on the set A A of all positive atoms in E E of norm one and λ a {lambda }_{a} is given by T A a = λ a a {T}_{A}a={lambda }_{a}a for all a ∈ A ain A .
设E E为复巴拿赫格，T T为中心Z (E) =上的算子 { T:∣T∣≤λ I对于某些λ } zleft(e)=left｛t:| t | le lambda Ihspace{0.33em}hspace{0.1em}text{for some}hspace{0.1em}hspace{0.33em}lambda right｝ E E。然后，基本规范‖T‖e Vert t{Vert }＿{e} T =本质谱半径r e (T) {r}＿{e}left(T) (T)我们也证明了re (T) = max { ‖T A d‖，r e (T A) } {r}＿{e}left(t)=max left｛Vert {t}＿{}hspace{-0.35em}{}＿{{a}＾{d}}Vert ，{r}＿{e}left（{t}＿{a}）right｝，其中T {t}＿{a} T的原子部分是T还是T是d {t}＿{}hspace{-0.35em}{}＿{{a}＾{d}} 是T T的非原子部分。并且，re (T A) = limsup λ A {r}＿{e}left（{t}＿{a}）={mathrm{limsup}}＿{{mathcal{ {mathcal F} }}}{lambda }＿{a} ，其中: {mathcal{ {mathcal F} }} fr过滤器是在集合A A上的吗?集合A A是E E的范数1和λ A的所有正原子 {lambda }＿{a} 由T A A = λ A A给出 {t}＿{a}a={lambda }＿{a}对于所有的a∈ain 选A。
{"title":"The essential spectrum, norm, and spectral radius of abstract multiplication operators","authors":"A. R. Schep","doi":"10.1515/conop-2022-0141","DOIUrl":"https://doi.org/10.1515/conop-2022-0141","url":null,"abstract":"Abstract Let E E be a complex Banach lattice and T T is an operator in the center Z ( E ) = { T : ∣ T ∣ ≤ λ I for some λ } Zleft(E)=left{T:| T| le lambda Ihspace{0.33em}hspace{0.1em}text{for some}hspace{0.1em}hspace{0.33em}lambda right} of E E . Then, the essential norm ‖ T ‖ e Vert T{Vert }_{e} of T T equals the essential spectral radius r e ( T ) {r}_{e}left(T) of T T . We also prove r e ( T ) = max { ‖ T A d ‖ , r e ( T A ) } {r}_{e}left(T)=max left{Vert {T}_{}hspace{-0.35em}{}_{{A}^{d}}Vert ,{r}_{e}left({T}_{A})right} , where T A {T}_{A} is the atomic part of T T and T A d {T}_{}hspace{-0.35em}{}_{{A}^{d}} is the nonatomic part of T T . Moreover, r e ( T A ) = limsup ℱ λ a {r}_{e}left({T}_{A})={mathrm{limsup}}_{{mathcal{ {mathcal F} }}}{lambda }_{a} , where ℱ {mathcal{ {mathcal F} }} is the Fréchet filter on the set A A of all positive atoms in E E of norm one and λ a {lambda }_{a} is given by T A a = λ a a {T}_{A}a={lambda }_{a}a for all a ∈ A ain A .","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"10 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43629578","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 Let (X, 𝒜, μ) be a σ−finite measure space. A transformation ϕ : X → X is non-singular if μ ∘ ϕ−1 is absolutely continuous with respect with μ. For this non-singular transformation, the composition operator Cϕ: 𝒟(Cϕ) → L2(μ) is defined by Cϕf = f ∘ ϕ, f ∈ 𝒟(Cϕ). For a fixed positive integer n ≥ 2, basic properties of product Cϕn · · · Cϕ1 in L2(μ) are presented in Section 2, including the boundedness and adjoint. Under the assistance of these properties, normality and quasinormality of specific bounded Cϕn · · · Cϕ1 in L2(μ) are characterized in Section 3 and 4 respectively, where Cϕ1, Cϕ2, · · ·, Cϕn are all densely defined.
{"title":"Normality and Quasinormality of Specific Bounded Product of Densely Defined Composition Operators in L2 Spaces","authors":"Hang Zhou","doi":"10.1515/conop-2022-0130","DOIUrl":"https://doi.org/10.1515/conop-2022-0130","url":null,"abstract":"Abstract Let (X, 𝒜, μ) be a σ−finite measure space. A transformation ϕ : X → X is non-singular if μ ∘ ϕ−1 is absolutely continuous with respect with μ. For this non-singular transformation, the composition operator Cϕ: 𝒟(Cϕ) → L2(μ) is defined by Cϕf = f ∘ ϕ, f ∈ 𝒟(Cϕ). For a fixed positive integer n ≥ 2, basic properties of product Cϕn · · · Cϕ1 in L2(μ) are presented in Section 2, including the boundedness and adjoint. Under the assistance of these properties, normality and quasinormality of specific bounded Cϕn · · · Cϕ1 in L2(μ) are characterized in Section 3 and 4 respectively, where Cϕ1, Cϕ2, · · ·, Cϕn are all densely defined.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"86 - 95"},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41909001","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 Operators of type f → ψf ◦ φ acting on function spaces are called weighted composition operators. If the weight function ψ is the constant function 1, then they are called composition operators. We consider weighted composition operators acting on Hardy–Smirnov spaces and prove that their unitarily invariant properties are reducible to the study of weighted composition operators on the classical Hardy space over a disc. We give examples of such results, for instance proving that Forelli’s theorem saying that the isometries of non–Hilbert Hardy spaces over the unit disc need to be special weighted composition operators extends to all non–Hilbert Hardy–Smirnov spaces. A thorough study of boundedness of weighted composition operators is performed.
{"title":"Weighted composition operators on Hardy–Smirnov spaces","authors":"Valentin Matache","doi":"10.1515/conop-2022-0136","DOIUrl":"https://doi.org/10.1515/conop-2022-0136","url":null,"abstract":"Abstract Operators of type f → ψf ◦ φ acting on function spaces are called weighted composition operators. If the weight function ψ is the constant function 1, then they are called composition operators. We consider weighted composition operators acting on Hardy–Smirnov spaces and prove that their unitarily invariant properties are reducible to the study of weighted composition operators on the classical Hardy space over a disc. We give examples of such results, for instance proving that Forelli’s theorem saying that the isometries of non–Hilbert Hardy spaces over the unit disc need to be special weighted composition operators extends to all non–Hilbert Hardy–Smirnov spaces. A thorough study of boundedness of weighted composition operators is performed.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"160 - 176"},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47454508","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 Let A be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry, V, so that AV > 0 and A is Hankel with respect to V, i.e. V*A = AV, if and only if A is not invertible. The isometry V can be chosen to be isomorphic to N ∈ ℕ ∪ {+∞} copies of the unilateral shift if A has spectral multiplicity at most N. We further show that the set of all isometries, V, so that A is Hankel with respect to V, are in bijection with the set of all closed, symmetric restrictions of A−1.
设A是复可分Hilbert空间上的一个有界、内射、自伴随线性算子。我们证明存在一个纯等距，V，使得AV > 0和a是关于V的Hankel，即V* a = AV，当且仅当a不可逆。如果A的谱多重性不超过N，则可以选择等距V同构于N∈N∪{+∞}的单侧位移副本。我们进一步证明了使A相对于V是Hankel的所有等距集合V与A−1的所有闭对称限制集合双射。
{"title":"On unitary equivalence to a self-adjoint or doubly–positive Hankel operator","authors":"R.T.W. Martin","doi":"10.1515/conop-2022-0132","DOIUrl":"https://doi.org/10.1515/conop-2022-0132","url":null,"abstract":"Abstract Let A be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry, V, so that AV > 0 and A is Hankel with respect to V, i.e. V*A = AV, if and only if A is not invertible. The isometry V can be chosen to be isomorphic to N ∈ ℕ ∪ {+∞} copies of the unilateral shift if A has spectral multiplicity at most N. We further show that the set of all isometries, V, so that A is Hankel with respect to V, are in bijection with the set of all closed, symmetric restrictions of A−1.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"114 - 126"},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43526464","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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