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.