On a class of multiparameter perturbations of positive definite operators with fixed bounds on their spectrum (Q1386505)

Let \(H\) be a Hilbert space with inner product \((\cdot,\cdot)\). For \(0 < a < b\) denote by \(S_H\{a,b\}\) the set of all bounded symmetric operators in \(H\) such that \[ a = \inf_{\| f\| =1} (Af,f), \qquad b = \sup_{\| f\| =1} (Af,f). \] The main result of the paper is the following Theorem. Let \(\alpha_j\in (0,1]\), \(j=1,\ldots,m\), and let \(P_j\), \(j=1,\ldots,m\), pairwise orthogonal orthoprojections in \(H\). Then \[ A + \sum_{j=1}^m (\alpha_j-1)P_j AP_j \] is positive definite for each \(A\in S_H \{a,b\}\) if \[ \alpha_1 \ldots \alpha_m > \left( b-a \over b+a \right)^2. \tag{\(*\)} \] For \(m=1\) and \(m=2\) it is proved that the condition \((*)\) is also necessary. The result for \(m=1\) goes back to a paper by A. S. Markus and S. D. Eidel'man in 1970.