Hendrik Schlieter, Wend Werner
We introduce unbounded multipliers on operator spaces. These multipliers generalize both, regular operators on Hilbert C*-modules and (bounded) multipliers on operator spaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We introduce unbounded multipliers on operator spaces. These multipliers generalize both, regular operators on Hilbert C*-modules and (bounded) multipliers on operator spaces. ----- Wir definieren den Begriff eines unbeschränkten Multiplikators auf Operatorräumen. Diese Multiplikatoren verallgemeinern sowohl reguläre Operatoren auf Hilbert-C*-Moduln als auch (beschränkte) Multiplikatoren auf Operatorräumen.
A regular symmetric operator on a Hilbert module is self-adjoint whenever there exists a suitable approximate identity. We say an operator is 'locally bounded' if the composition of the operator with each element in the approximate identity is bounded. We prove that the perturbation of a regular self-adjoint operator by a locally bounded symmetric operator is again regular and self-adjoint. We use this result to show that the Kasparov class represented by an unbounded Kasparov module is stable under locally bounded perturbations. As an application, we show that we obtain a converse to the 'dou…
In this paper, we introduce the notion of multiplier of a Hilbert algebra. The space of bounded multipliers is a semifinite von Neumann algebra isomorphic to the left von Neumann algebra of the Hilbert algebra, as expected. However, in the unbounded setting, the space of multipliers has the structure of a *-algebra with nice properties concerning commutant and affiliation: it is a pre-GW*-algebra. And this correspondence between Hilbert algebras and its multipliers is functorial. Then, we can endow the Hilbert algebra with a nice topology constructed from unbounded multipliers. As we can see f…
In a recent survey paper we introduced one-sided multipliers between two different operator spaces. Here we give some basic theory for these maps.
It is known that the generator of a strictly continuous one parameter unitary group in the multiplier algebra of a $C^{*}$-algebra is affiliated to that $C^{*}$-algebra . We show that under natural non degeneracy conditions, this self adjoint unbounded operator lies indeed in the (unbounded) multiplier algebra of the Pedersen's ideal of the $C^{*}$-algebra.
We state several equivalent noncommutative versions of the Cauchy-Riemann equations and characterize the unbounded operators on L^2(R) which satisfy them. These operators arise from the creation operator via a functional calculus involving a class of entire functions, identified by Newman and Shapiro [D. J. Newman and H. S. Shapiro, Fischer spaces of entire functions, in Entire Functions and Related Parts of Analysis (J. Koorevaar, ed.), AMS Proc. Symp. Pure Math. XI (1968), 360-369], which act as unbounded multiplication operators on Bargmann-Segal space.
We continue the study of multidimensional operator multipliers initiated in [arXiv:math/0701645]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C*-algebras is automatically compact. We give a description of multilinear modular completely com…
We show that some matrices are Schur multipliers and this is applied to obtain classes of operator-valued Foguel-Hankel operators similar to contractions. This provides partial answers to a problem of K. Davidson and the second author concerning CAR-valued Foguel-Hankel operators.
Schur multipliers were introduced by Schur in the early 20th century and have since then found a considerable number of applications in Analysis and enjoyed an intensive development. Apart from the beauty of the subject in itself, sources of interest in them were connections with Perturbation Theory, Harmonic Analysis, the Theory of Operator Integrals and others. Advances in the quantisation of Schur multipliers were recently made by Kissin and Shulman. The aim of the present article is to summarise a part of the ideas and results in the theory of Schur and operator multipliers. We start with …
We introduce multidimensional Schur multipliers and characterise them generalising well known results by Grothendieck and Peller. We define a multidimensional version of the two dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several C*-algebras satisfying certain boundedness conditions. In the case of commutative C*-algebras, the multidimensional operator multipliers reduce to continuous multidimensional Schur multipliers. We show that the multipliers with respect to som…
Let $A$ and $B$ be two densely defined unbounded closeable operators in a Hilbert space such that their unbounded operator products $AB$ and $BA$ are also densely defined. Then all four operators possess adjoints and we obtain new inclusion bounds for the operator product closures $\bar{A} \bar{B}$ and $\bar{AB}$ in terms of new relations among the operator adjoints. These in turn lead to sharpened understandings for when products of unbounded self-adjoint and unbounded normal operators are self-adjoint and normal. They also clarify certain operator-product issues for Dirac operators.
The theory of one-sided $M$-ideals and multipliers of operator spaces is simultaneously a generalization of classical $M$-ideals, ideals in operator algebras, and aspects of the theory of Hilbert $C^*$-modules and their maps. Here we give a systematic exposition of this theory; a reference tool for `noncommutative functional analysts' who may encounter a one-sided $M$-ideal or multiplier in their work.
Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the c…
In this note, we give the most natural (perhaps the simplest ever) generalization of the Fuglede-Putnam theorem where all operators involved are unbounded.
Affiliated and normal operators in octonion Hilbert spaces are studied. Theorems about their properties and of related algebras are demonstrated. Spectra of unbounded normal operators are investigated.
In the present paper we give results on the closedness and the self-adjointness of the sum of two unbounded operators. We present a new approach to these fundamental questions in operator theory. We also prove a new version of the Fuglede theorem where the operators involved are all unbounded. This new Fuglede theorem allows us to prove (under extra conditions) that the sum of two unbounded normal operators remains normal. Also a result on the normality of the unbounded product of two normal operators is obtained as a consequence of this new "Fuglede theorem". Some interesting examples are als…
We investigate a limiting procedure for extending local integral operator equalities to the global ones and to applying it to obtaining generalizations of the Newton-Leibnitz formula for operator-valued maps for a wide class of unbounded operators.
For a collection of reproducing kernels k which includes those for the Hardy space of the polydisk and ball and for the Bergman space, k is a complete Pick kernel if and only if the multiplier algebra of the Hilbert space H^2(k) associated to k has the Douglas property. Consequences for solving the operator equation AX=Y are examined.
The present paper is exclusively devoted to counterexamples about commutators and self commutators of unbounded operators on a Hilbert space. As a bonus, we provide a simpler counterexample than McIntosh's famous example obtained some while ago.
In this paper, we establish results about operators similar to their adjoints. This is carried out in the setting of bounded and also unbounded operators on a Hilbert space. Among the results, we prove that an unbounded closed operator similar to its adjoint, via a cramped unitary operator, is self-adjoint. The proof of this result works also as a new proof of the celebrated result by Berberian on the same problem in the bounded case. Other results on similarity of hyponormal unbounded operators and their self-adjointness are also given, generalizing famous results by Sheth and Williams.
In this paper, we are mainly concerned with studying arbitrary unbounded square roots of linear operators as well as some of their basic properties. The paper contains many examples and counterexamples. As an illustration, we give explicit everywhere defined unbounded non-closable $nth$ roots of the identity operator as well as the zero operator. We also show a non-closable unbounded operator without any non-closable square root. Among other consequences, we have a way of finding everywhere defined bijective operators, everywhere defined operators which are surjective without being injective a…
Devinatz, Nussbaum and von Neumann established some important results on the strong commutativity of self-adjoint and normal unbounded operators. In this paper, we prove results in the same spirit.
On unbounded complementable operators
本文扩展了可补性概念到无界算子，通过投影和分解技术分析其结构，为算子理论的发展提供了新见解。
In this paper we give and prove a criterion for the normality of unbounded closed operators, which is a sort of a maximality result which will be called "double maximality". As applications, we show, under some assumptions, that the sum of two symmetric operators is essentially self-adjoint; and that the sum of two unbounded normal operators is essentially normal. Some other important results are also established.
We establish an unbounded version of Stinespring's Theorem and a lifting result for Stinespring representations of completely positive modular maps defined on the space of all compact operators. We apply these results to study positivity for Schur multipliers. We characterise positive local Schur multipliers, and provide a description of positive local Schur multipliers of Toeplitz type. We introduce local operator multipliers as a non-commutative analogue of local Schur multipliers, and obtain a characterisation that extends earlier results concerning operator multipliers and local Schur mult…
In this paper, we give an example of a closed unbounded operator whose square's domain and adjoint's square domain are equal and trivial. Then, we come up with an essentially self-adjoint whose square has a trivial domain.
A result of Gilbert shows that every completely bounded multiplier $f$ of the Fourier algebra $A(G)$ arises from a pair of bounded continuous maps $α,β:G \rightarrow K$, where $K$ is a Hilbert space, and $f(s^{-1}t) = (β(t)|α(s))$ for all $s,t\in G$. We recast this in terms of adjointable operators acting between certain Hilbert C$^*$-modules, and show that an analogous construction works for completely bounded left multipliers of a locally compact quantum group. We find various ways to deal with right multipliers: one of these involves looking at the opposite quantum group, and this leads to …
This paper has been withdrawn by the author due to a crucial sign error in equation 1.
Some properties of unbounded truncated Toeplitz operators
本文研究了在模型空间上由内函数u定义的闭稠密定义的无界截断Toeplitz算子，研究其与修改压缩位移算子的交换性及可逆性和自伴性性质。
The theory of M-ideals and multiplier mappings of Banach spaces naturally generalizes to left (or right) M-ideals and multiplier mappings of operator spaces. These subspaces and mappings are intrinsically characterized in terms of the matrix norms. In turn this is used to prove that the algebra of left adjointable mappings of a dual operator space X is a von Neumann algebra. If in addition X is an operator A--B-bimodule for $C^{*}$-algebras A and B, then the module operations on X are automatically weak$^{*}$ continuous. One sided L-projections are introduced, and analogues of various results …
In this paper we deal with unbounded composition operators defined in Orlicz spaces. Indeed, we provide some necessary and sufficient condition for densely definedness of composition operators on Orlicz spaces. Also, we will investigate the adjoint of densely defined composition operators and we give some equivalent conditions for it to be densely defined. In addition, we show that densely defined composition operator is continuous if and only if it is everywhere defined. Finally, we characterize densely defined continuous composition operators.
We generalize some technical results of Glicksberg to the realm of general operator algebras and use them to give a characterization of open and closed projections in terms of certain multiplier algebras. This generalizes a theorem of J. Wells characterizing an important class of ideals in uniform algebras. The difficult implication in our main theorem is that if a projection is open in an operator algebra, then the multiplier algebra of the associated hereditary subalgebra arises as the closure of the subalgebra with respect to the strict topology of the multiplier algebra of a naturally asso…
The paper contains results on the structure of Jordan maps and several kinds of triple maps on standard algebras of unbounded operators in Hilbert spaces. These results are unbounded counterparts to results on algebras of bounded operators obtained by Lu, Molnár and others ([3,4,6 -- 10]).
Applying a result of abstract ring theory we get that bijective additive mappings on standard algebras of unbounded operators preserving zero products are multiples of ring isomorphisms. The structure of additive bijective mappings on certain classes of standard algebras of unbounded operators preserving orthogonality in both directions is also investigated. The results are quite similar to those for algebras of bounded operators
For a given C*-algebra $\mathcal{A}$, we establish the existence of maximal and minimal operator $\mathcal{A}$-system structures on an AOU $\mathcal{A}$-space. In the case $\mathcal{A}$ is a W*-algebra, we provide an abstract characterisation of dual operator $\mathcal{A}$-systems, and study the maximal and minimal dual operator $\mathcal{A}$-system structures on a dual AOU $\mathcal{A}$-space. We introduce operator-valued Schur multipliers, and provide a Grothendieck-type characterisation. We study the positive extension problem for a partially defined operator-valued Schur multiplier $φ$ and…
The impact of Schur multipliers in harmonic analysis and operator algebras
本文综述舒尔乘子在过去15年的影响，关注其与傅里叶乘子的联系及在调和分析和算子代数间的桥梁作用，介绍舒尔滕p类最新界在相关领域的应用，以及非托普利茨舒尔乘子等方面的新进展。
Resolvents of quasi-linear operators and operator algebras in Banach spaces over the quaternion field are investigated. Spectral theory of unbounded nonlinear operators in quaternion Banach spaces is studied. Strongly continuous semigroups of quaternion quasi-linear operators are investigated and the polar decomposition theorem for them is proved.
Representations of Hermitian Commutative *-Algebras by Unbounded Operators
本文针对埃尔米特交换幺 *-代数，证明了由预希尔伯特空间中可能无界的算子给出的幺表示的谱定理，并指出可数生成情形下已有更优结果。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。