On trace and Hilbert–Schmidt norm estimates

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.