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On trace and Hilbert–Schmidt norm estimates

On trace and Hilbert–Schmidt norm estimates Let ℰ and 풫 be nonnegative quadratic forms in the Hilbert space ℋ. Suppose that, for every β ⩾ 0, the form ℰ+β풫 is densely defined and closed. Let Hβ be the self‐adjoint operator associated with ℰ+β 풫 and R∞ := lim β→∞ (Hβ+1)−1. We give estimates for the distance between (Hβ+1)−1 and R∞ with respect to the norm ‖·‖p in the Schatten–von Neumann class of order p, p=1, 2. In particular, we derive a condition that is necessary and sufficient in order that ‖(Hβ + 1)−1 − R∞‖1⩽ c/β∀β > 0 for some finite constant c, and give examples where this criterion is satisfied. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On trace and Hilbert–Schmidt norm estimates

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References (17)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/bdr131
Publisher site
See Article on Publisher Site

Abstract

Let ℰ and 풫 be nonnegative quadratic forms in the Hilbert space ℋ. Suppose that, for every β ⩾ 0, the form ℰ+β풫 is densely defined and closed. Let Hβ be the self‐adjoint operator associated with ℰ+β 풫 and R∞ := lim β→∞ (Hβ+1)−1. We give estimates for the distance between (Hβ+1)−1 and R∞ with respect to the norm ‖·‖p in the Schatten–von Neumann class of order p, p=1, 2. In particular, we derive a condition that is necessary and sufficient in order that ‖(Hβ + 1)−1 − R∞‖1⩽ c/β∀β > 0 for some finite constant c, and give examples where this criterion is satisfied.

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Aug 1, 2012

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