On the extension of bimeasures

Karni, Shaul; Merzbach, Ely

doi:10.1007/bf02789194

Cited by 11 publications

(9 citation statements)

References 12 publications

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“…It is enough (see, e.g., [14,Theorem 2.12] for p = 2) to prove that the variation of the set function m on the rectangles of R p is bounded by V 1 2 V 2 2 · · · V p 2 , which can be accomplished completely analogously to the proof of [20, Theorem 1] (see also [5]). 2…”

Section: Multiple Spectral Measuresmentioning

confidence: 97%

Higher order spectral shift

Dykema

Skripka

2009

Journal of Functional Analysis

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We construct higher order spectral shift functions, extending the perturbation theory results of M.G. Krein [M.G. Krein, On a trace formula in perturbation theory, Mat. Sb. 33 (1953) 597-626 (in Russian)] and L.S. Koplienko [L.S. Koplienko, Trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (1984) 62-71 (in Russian); translation in: Trace formula for nontrace-class perturbations, Siberian Math. J. 25 (1984) 735-743] on representations for the remainders of the first and second order Taylor-type approximations of operator functions. The higher order spectral shift functions represent the remainders of higher order Taylor-type approximations; they can be expressed recursively via the lower order (in particular, Krein's and Koplienko's) ones. We also obtain higher order spectral averaging formulas generalizing the Birman-Solomyak spectral averaging formula. The results are obtained in the semi-finite von Neumann algebra setting, with the perturbation taken in the Hilbert-Schmidt class of the algebra.

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Section: Multiple Spectral Measuresmentioning

confidence: 97%

Higher order spectral shift

Dykema

Skripka

2009

Journal of Functional Analysis

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“…As mentioned there, the result was proven for the special case of bilinear forms in [16], but it does not seem possible to extend the proof techniques of [16] to m > 2. Then the following are equivalent:…”

Section: Theorem 21 ([2]mentioning

confidence: 99%

“…The answer to this for the case of Radon polymeasures appeared for the first time in [2], although the particular case of bimeasures, or bilinear operators, had already been studied in [16]. The answer for bounded additive polymeasures is simpler and follows along the same lines.…”

Section: Introductionmentioning

confidence: 99%

Characterization of dual mixed volumes via polymeasures

Jimenez

Villanueva

2015

Journal of Mathematical Analysis and Applications

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“…A major example is given by the work [8] by Grothendieck, where he introduced what is now called the Grothendieck's inequality, namely what he described as "the fundamental theorem in the metric theory of tensor products". Further relevant works, which include also the efforts of obtaining the Riesz-Markov-Kakutani representation theorem in the bilinear and multilinear case, are [1,2,3,6,9,11,18,21,22], among others.…”

Section: Introductionmentioning

confidence: 99%

On the extension and kernels of signed bimeasures and their role in stochastic integration

Passeggeri¹

2020

Preprint

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In this work we provide a necessary and sufficient condition for the extension of signed bimeasures on δ-rings and for the existence of relative kernels. This result generalises the construction method of regular conditional probabilities to the more general setting of extended signed measures. Building on this result, we obtain the most general theory of stochastic integrals based on random measures, thus extending and generalising the whole integration theory developed in the celebrated Rajput and Rosinski's paper (Probab. Theory Relat. Fields, 82 (1989) 451-487) and the recent results by Passeggeri (Stoch. Process. Their Appl., 130, (3), (2020), 1735-1791).

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On the extension of bimeasures

Cited by 11 publications

References 12 publications

Higher order spectral shift

Higher order spectral shift

Characterization of dual mixed volumes via polymeasures

On the extension and kernels of signed bimeasures and their role in stochastic integration

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