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On trace and Hilbert–Schmidt norm estimates
Author(s) -
BelHadjAli H.,
BenAmor A.,
Brasche J.
Publication year - 2012
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdr131
Subject(s) - mathematics , hilbert space , norm (philosophy) , quadratic equation , trace class , combinatorics , order (exchange) , trace (psycholinguistics) , pure mathematics , operator norm , mathematical analysis , geometry , linguistics , philosophy , finance , political science , law , economics
Let ℰ and be nonnegative quadratic forms in the Hilbert space ℋ . Suppose that, for every β ⩾ 0, the form ℰ + β is densely defined and closed. Let H β be the self‐adjoint operator associated with ℰ + β and R ∞ := lim β →∞ ( H β +1) −1 . We give estimates for the distance between ( H β +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‖( H β + 1 )− 1 − R ∞ ‖ 1 ⩽ c / β∀ β > 0for some finite constant c , and give examples where this criterion is satisfied.