Classes of Operator-Smooth Functions. Iii. Stable Functions and Fuglede Ideals

Kissin, Edward; Shulman, Victor S.

doi:10.1017/s001309150300018x

Cited by 10 publications

(13 citation statements)

References 21 publications

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“…ideals. By Proposition 2.1 of [16], the norms · J and · J φ coincide on J φ 0 . Hence f is commutator J φ 0 -bounded on α.…”

Section: Hadamard Multipliersmentioning

confidence: 89%

“…We will show in [16] that, for a wide class of ideals J, all J-Lipschitz functions have this property.…”

Section: Operator Lipschitz Functionsmentioning

confidence: 93%

“…We prove that the spaces S 1 -CB(α), S ∞ -CB(α) and S b -CB(α) coincide and that the space S 2 -CB(α) consists of all functions on α that are Lipschitz in the usual sense. We use these results in [16] …”

Section: Introductionmentioning

confidence: 99%

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Classes of Operator-Smooth Functions. I. Operator-Lipschitz Functions

Kissin¹,

Shulman²

2005

Proceedings of the Edinburgh Mathematical Society

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In this paper we study the spaces of operator-Lipschitz functions and the spaces of functions closed to them: commutator bounded. Apart from the standard operator norm on B(H), we consider a rich variety of symmetric operator norms and spaces of operator-Lipschitz functions with respect to these norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of operator-Lipschitz functions.

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“…ideals. By Proposition 2.1 of [16], the norms · J and · J φ coincide on J φ 0 . Hence f is commutator J φ 0 -bounded on α.…”

Section: Hadamard Multipliersmentioning

confidence: 89%

“…We will show in [16] that, for a wide class of ideals J, all J-Lipschitz functions have this property.…”

Section: Operator Lipschitz Functionsmentioning

confidence: 93%

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Classes of Operator-Smooth Functions. I. Operator-Lipschitz Functions

Kissin¹,

Shulman²

2005

Proceedings of the Edinburgh Mathematical Society

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“…Our main technical tool in the proof of this result is the theory of double operator integrals in the setting of symmetric operator spaces, recently developed in [12,13], and strongly inspired by the corresponding theory in the setting of symmetrically normed ideals of compact operators, which was developed by Birman and Solomyak in [3][4][5]. In the latter setting, classes of operator-smooth functions have been recently studied by Kissin and Shulman [22][23][24]. We briefly recall a few notions from this theory in § 5, below.…”

Section: Introductionmentioning

confidence: 99%

Commutator Estimates and $\Rr$-Flows in Non-Commutative Operator Spaces

Pagter¹,

Sukochev²

2007

Proceedings of the Edinburgh Mathematical Society

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The principal results in this paper are concerned with the description of domains of infinitesimal generators of strongly continuous groups of isometries in non-commutative operator spaces E(M, τ), which are induced by R-flows on M. In particular, we are concerned with the description of operator functions which leave the domain of such generators invariant in all symmetric operator spaces, associated with a semi-finite von Neumann algebra M and a separable function space E on (0, ∞). Furthermore, we apply our results to the study of operator functions for which [D, x] , where D is an unbounded self-adjoint operator. Our methods are partly based on the recently developed theory of double operator integrals in symmetric operator spaces and the theory of adjoint C 0 -semigroups.

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“…ideals and J ⊆ I , Proposition 2.1 of [Kissin and Shulman 2005b] tells us that there is c > 0 such that…”

Section: Preliminariesmentioning

confidence: 99%

Operator multipliers

Kissin

Shulman

2006

Pacific J. Math.

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We introduce a noncommutative version of Schur multipliers relative to an operator ideal. In this setting the functions of two variables are replaced by elements from a tensor product of C*-algebras, and the measures (or spectral measures) by representations. For commutative C*-algebras this approach agrees with Birman and Solomyak's theory of double operator integrals. We study the dependence of the spaces of multipliers on the choice of representations and find that the question is closely related to Voiculescu and Arveson's theory of approximately equivalent representations. The space of multipliers universal with respect to the chosen measures is related to the Haagerup tensor product of the algebras.

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Classes of Operator-Smooth Functions. Iii. Stable Functions and Fuglede Ideals

Cited by 10 publications

References 21 publications

Classes of Operator-Smooth Functions. I. Operator-Lipschitz Functions

Classes of Operator-Smooth Functions. I. Operator-Lipschitz Functions

Commutator Estimates and $\Rr$-Flows in Non-Commutative Operator Spaces

Operator multipliers

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