On trace and Hilbert-Schmidt norm estimates (Q2903269)

The paper under review is devoted to the discussion of problems related to various types of large coupling convergence in abstract Hilbert spaces. Let \(\mathcal E\) and \(P\) be nonnegative quadratic forms in the Hilbert space \(\mathcal H\). Suppose that, for every \(\beta \geq 0\), the form \({\mathcal E} + \beta P\) is densely defined and closed. Let \(H_\beta\) be the self adjoint operator associated with \({\mathcal E} + \beta P\) and \(R_\infty := \lim\limits_{\beta\to\infty}(H_\beta + 1)^{-1}\) (in the strong operator topology). The authors give estimates for the distance between \((H_\beta + 1)^{-1}\) and \(R_\infty\) with respect to the norm \(\|\cdot\|_p\) in the Schatten-von Neumann class of order \(p\), \(p = 1, 2\). In particular, the authors derive a condition that is necessary and sufficient in order that NEWLINE\[NEWLINE \|(H_\beta + 1)^{-1} - R_\infty \|_1 \leq c/\beta \quad\mathrm{for all}\,\, \beta > 0NEWLINE\]NEWLINE for some finite constant \(c\), and give examples where this criterion is satisfied. Moreover, conditions that are sufficient for the estimate \( \|(H_\beta + 1)^{-1} - R_\infty \|_p \leq c/\beta^r\) are obtained.