. We introduce two new concepts for unbounded operators T in a Hilbert space, the essential numerical range W e 5 ( T ) of type 5 and the C -numerical range W C ( T ) . Our first main result clarifies the relation of W e 5 ( T ) to the essential numerical range W e ( T ) , answering an open problem of Bögli, Marletta and Tretter’s (2020) by employing the Bessaga–Pełczyński selection theorem from Banach space theory. It turns out that W e 5 ( T ) ⊂ W e ( T ) and we establish sharp conditions for equality. An example for strict inclusion shows that W e ( T ) may be a half-plane, while W e 5 ( T ) only a line. We also show that W e 5 ( T ) is convex and that it contains the convex hull of the essential spectrum. Our second main result reveals a geometric relation between W e 5 ( T ) and W C ( T ) . We show that, for finite-rank operators C , W C ( T ) is star-shaped with star-centre (Tr C ) W e 5 ( T ) , generalizing a result for bounded operators where W e 5 ( T ) = W e ( T ) .
． 我们引入了Hilbert空间中无界算子T的两个新概念，即5型的基本数值范围w5 (T)和C -数值范围wc (T)。我们的第一个主要结果阐明了we 5 (T)与基本数值范围we (T)的关系，通过使用巴拿赫空间理论中的Bessaga-Pełczyński选择定理回答了Bögli, Marletta和Tretter(2020)的一个开放问题。结果是we 5 (T)∧W e (T)，我们建立了相等的尖锐条件。一个严格包含的例子表明，we (T)可能是半平面，而we5 (T)只是一条直线。我们还证明了we5 (T)是凸的，并且它包含了基本谱的凸包。我们的第二个主要结果揭示了w5 (T)和wc (T)之间的几何关系。我们证明了对于有限秩算子C, wc (T)是星中心(Tr C) We 5 (T)的星形，推广了We 5 (T) = We (T)的有界算子的结果。
{"title":"Essential numerical range and $C$-numerical range\u0000for unbounded operators","authors":"N. Hefti, C. Tretter","doi":"10.4064/sm201231-16-9","DOIUrl":"https://doi.org/10.4064/sm201231-16-9","url":null,"abstract":". We introduce two new concepts for unbounded operators T in a Hilbert space, the essential numerical range W e 5 ( T ) of type 5 and the C -numerical range W C ( T ) . Our first main result clarifies the relation of W e 5 ( T ) to the essential numerical range W e ( T ) , answering an open problem of Bögli, Marletta and Tretter’s (2020) by employing the Bessaga–Pełczyński selection theorem from Banach space theory. It turns out that W e 5 ( T ) ⊂ W e ( T ) and we establish sharp conditions for equality. An example for strict inclusion shows that W e ( T ) may be a half-plane, while W e 5 ( T ) only a line. We also show that W e 5 ( T ) is convex and that it contains the convex hull of the essential spectrum. Our second main result reveals a geometric relation between W e 5 ( T ) and W C ( T ) . We show that, for finite-rank operators C , W C ( T ) is star-shaped with star-centre (Tr C ) W e 5 ( T ) , generalizing a result for bounded operators where W e 5 ( T ) = W e ( T ) .","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70509098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Trotter–Kato product formula in symmetric F-normed ideals","authors":"Meiram Akhymbek, G. Levitina","doi":"10.4064/sm210708-4-11","DOIUrl":"https://doi.org/10.4064/sm210708-4-11","url":null,"abstract":"","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70515664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Kato’s inequality for the strong $p(cdot)$-Laplacian","authors":"T. Do, L. Truong","doi":"10.4064/sm220330-19-9","DOIUrl":"https://doi.org/10.4064/sm220330-19-9","url":null,"abstract":"","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70526573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract: The paper is devoted to the convex-set counterpart of the theory of weak derived sets initiated by Banach and Mazurkiewicz for subspaces. The main result is the following: For every nonreflexive Banach space X and every countable successor ordinal α, there exists a convex subset A in X such that α is the least ordinal for which the weak derived set of order α coincides with the weak closure of A. This result extends the previously known results on weak derived sets by Ostrovskii (2011) and Silber (2021).
{"title":"Weak$^*$ closures and derived sets for convex sets in dual Banach spaces","authors":"M. Ostrovskii","doi":"10.4064/sm211211-25-6","DOIUrl":"https://doi.org/10.4064/sm211211-25-6","url":null,"abstract":"Abstract: The paper is devoted to the convex-set counterpart of the theory of weak derived sets initiated by Banach and Mazurkiewicz for subspaces. The main result is the following: For every nonreflexive Banach space X and every countable successor ordinal α, there exists a convex subset A in X such that α is the least ordinal for which the weak derived set of order α coincides with the weak closure of A. This result extends the previously known results on weak derived sets by Ostrovskii (2011) and Silber (2021).","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46726858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We investigate the problem of existence of a bounded extension to C ( K ) of a bounded c 0 ( I )-valued operator T defined on the subalgebra of C ( K ) induced by a continuous increasing surjection φ : K → L , where K and L are compact lines. Generalizations of some of the results of [6] about extension of c 0 -valued operators are obtained. For instance, we prove that when a bounded extension of T exists then an extension can be obtained with norm at most twice the norm of T . Moreover, the class of compact lines L for which the c 0 -extension property is equivalent to the c 0 ( I )-extension property for any continuous increasing surjection φ : K → L is studied.
{"title":"Extension of $c_0(I)$-valued operators on spaces of\u0000continuous functions on compact lines","authors":"Victor dos Santos Ronchim, D. Tausk","doi":"10.4064/sm211120-2-6","DOIUrl":"https://doi.org/10.4064/sm211120-2-6","url":null,"abstract":". We investigate the problem of existence of a bounded extension to C ( K ) of a bounded c 0 ( I )-valued operator T defined on the subalgebra of C ( K ) induced by a continuous increasing surjection φ : K → L , where K and L are compact lines. Generalizations of some of the results of [6] about extension of c 0 -valued operators are obtained. For instance, we prove that when a bounded extension of T exists then an extension can be obtained with norm at most twice the norm of T . Moreover, the class of compact lines L for which the c 0 -extension property is equivalent to the c 0 ( I )-extension property for any continuous increasing surjection φ : K → L is studied.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46673693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. In 1964 Lochs proved a theorem on the number of continued fraction digits of a real number x that can be determined from just knowing its first n decimal digits. In 2001 this result was generalised to a dynamical systems setting by Dajani and Fieldsteel, where it compares sizes of cylinder sets for different transformations. In this article we prove a version of Lochs’ Theorem for random dynamical systems as well as a corresponding Central Limit Theorem. The main ingredient for the proof is an estimate on the asymptotic size of the cylinder sets of the random system in terms of the fiber entropy. To compute this entropy we provide a random version of Rokhlin’s formula for entropy.
{"title":"Random Lochs’ Theorem","authors":"Charlene Kalle, E. Verbitskiy, B. Zeegers","doi":"10.4064/sm211028-24-2","DOIUrl":"https://doi.org/10.4064/sm211028-24-2","url":null,"abstract":"Abstract. In 1964 Lochs proved a theorem on the number of continued fraction digits of a real number x that can be determined from just knowing its first n decimal digits. In 2001 this result was generalised to a dynamical systems setting by Dajani and Fieldsteel, where it compares sizes of cylinder sets for different transformations. In this article we prove a version of Lochs’ Theorem for random dynamical systems as well as a corresponding Central Limit Theorem. The main ingredient for the proof is an estimate on the asymptotic size of the cylinder sets of the random system in terms of the fiber entropy. To compute this entropy we provide a random version of Rokhlin’s formula for entropy.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let A be a Banach algebra and a, b ∈ A. The map Ma,b : A → A given by Ma,b(x) = axb is called a multiplication operator. Properties of compact multiplication operators have been investigated since 1964 when Vala published his work “On compact sets of compact operators” [15]. Let X be a normed space and B(X ) the algebra of all bounded linear maps from X into X . Vala proved that a nonzero multiplication operator Ma,b : B(X ) → B(X ) is compact if and only if the operators a, b ∈ B(X ) are both compact. Also, in [16] Vala defines an element a of a normed algebra to be compact if the mapping x 7→ axa is compact. This concept enabled the study of compactness properties of elements of abstract normed algebras. Ylinen in [17] studied compact elements for abstract C*-algebras and showed that a is a compact element of a C-algebra A if and only if there exists an isometric ∗-representation π of A on a Hilbert space H such that the operator π(a) is compact. Compactness questions have also been considered in the more general framework of elementary operators. A map Φ : A → A, where A is a Banach algebra, is called elementary if Φ = ∑m i=1 Mai,bi for some ai, bi ∈ A, i = 1, . . . ,m. Fong and Sourour showed that an elementary operator Φ : B(H) → B(H), where B(H) is the algebra of bounded linear operators on a Hilbert space H, is compact if and only if there exist compact operators ci, di ∈ B(H), i = 1, . . . ,m such that Φ = ∑m i=1 Mci,di [5]. This result was expanded by Mathieu on prime C*-algebras [9] and later on general C*-algebras by Timoney [14]. Akemann and Wright [1] characterized the weakly compact multiplication operators on B(H), where H is a Hilbert space. Saksman and Tylli [12, 13] and Johnson and Schechtman [6] studied weak compactness of multiplication operators in a Banach space setting. Moreover, strictly singular multiplication operators are studied by Lindström, Saksman and Tylli [8] and Mathieu and Tradacete [10]. Compactness properties of multiplication operators on nest algebras, a class of non selfadjoint operator algebras, are studied by Andreolas and Anoussis in [2]. In
设A是一个Banach代数，且A,b∈A。由Ma,b(x) = axb给出的映射Ma,b: A→A称为乘法算子。自从1964年Vala发表了他的著作“紧算子的紧集”[15]以来，紧乘法算子的性质就得到了研究。设X是赋范空间，B(X)是所有从X到X的有界线性映射的代数。Vala证明了非零乘法算子Ma,b: b (X)→b (X)是紧的当且仅当算子a,b∈b (X)都是紧的。同样，在[16]中，如果映射x7→axa是紧的，则Vala定义了范代数的元素a是紧的。这个概念使得研究抽象赋范代数元素的紧性成为可能。[17]中的Ylinen研究了抽象C*-代数的紧致元素，并证明了a是C-代数a的紧致元素，当且仅当a在Hilbert空间H上存在一个等距*表示π，使得算子π(a)紧致。紧性问题也在更一般的初等算子框架中被考虑过。一个映射Φ: A→A，其中A是一个Banach代数，如果Φ =∑m i=1 Mai,bi对于某些ai,bi∈A, i=1，…，则称为初等映射Φ: A→A。, m。Fong和Sourour证明了一个初等算子Φ: B(H)→B(H)，其中B(H)是Hilbert空间H上有界线性算子的代数，当且仅当存在紧算子ci, di∈B(H)， i = 1，…，m使得Φ =∑m i= 1mci,di[5]。这个结果由Mathieu在素数C*-代数[9]上推广，后来由Timoney在一般C*-代数[14]上推广。Akemann和Wright描述了B(H)上的弱紧乘法算子，其中H是Hilbert空间。Saksman and Tylli[12,13]和Johnson and Schechtman[10]研究了Banach空间下乘法算子的弱紧性。此外，Lindström、Saksman and Tylli[8]和Mathieu and Tradacete[10]研究了严格奇异乘法算子。Andreolas和Anoussis在[2]中研究了巢代数(一类非自伴算子代数)上乘法算子的紧性。在
{"title":"Compact multiplication operators on semicrossed products","authors":"G. Andreolas, M. Anoussis, C. Magiatis","doi":"10.4064/sm211107-22-7","DOIUrl":"https://doi.org/10.4064/sm211107-22-7","url":null,"abstract":"Let A be a Banach algebra and a, b ∈ A. The map Ma,b : A → A given by Ma,b(x) = axb is called a multiplication operator. Properties of compact multiplication operators have been investigated since 1964 when Vala published his work “On compact sets of compact operators” [15]. Let X be a normed space and B(X ) the algebra of all bounded linear maps from X into X . Vala proved that a nonzero multiplication operator Ma,b : B(X ) → B(X ) is compact if and only if the operators a, b ∈ B(X ) are both compact. Also, in [16] Vala defines an element a of a normed algebra to be compact if the mapping x 7→ axa is compact. This concept enabled the study of compactness properties of elements of abstract normed algebras. Ylinen in [17] studied compact elements for abstract C*-algebras and showed that a is a compact element of a C-algebra A if and only if there exists an isometric ∗-representation π of A on a Hilbert space H such that the operator π(a) is compact. Compactness questions have also been considered in the more general framework of elementary operators. A map Φ : A → A, where A is a Banach algebra, is called elementary if Φ = ∑m i=1 Mai,bi for some ai, bi ∈ A, i = 1, . . . ,m. Fong and Sourour showed that an elementary operator Φ : B(H) → B(H), where B(H) is the algebra of bounded linear operators on a Hilbert space H, is compact if and only if there exist compact operators ci, di ∈ B(H), i = 1, . . . ,m such that Φ = ∑m i=1 Mci,di [5]. This result was expanded by Mathieu on prime C*-algebras [9] and later on general C*-algebras by Timoney [14]. Akemann and Wright [1] characterized the weakly compact multiplication operators on B(H), where H is a Hilbert space. Saksman and Tylli [12, 13] and Johnson and Schechtman [6] studied weak compactness of multiplication operators in a Banach space setting. Moreover, strictly singular multiplication operators are studied by Lindström, Saksman and Tylli [8] and Mathieu and Tradacete [10]. Compactness properties of multiplication operators on nest algebras, a class of non selfadjoint operator algebras, are studied by Andreolas and Anoussis in [2]. In","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46054806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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