Compressions of linearly independent selfadjoint operators
For any positive integer \(k\) and for \(\mathbb{F}=\mathbb{R}\) or \(\mathbb{C}\), the authors consider the smallest number \(\gamma_{\mathbb{F}}(k)\) with the property that, for every \(k\) linearly independent selfadjoint operators \(X_1,\dots, X_k\) on a Hilbert space \(H\) over \(\mathbb{F}\), there is a subspace \(H_0\) of \(H\) for which the compressions \(X_j[H_0]\) of \(X_j\) to \(H_0\), \(1\leq j\leq k\), are still linearly independent. They first obtain lower and upper bounds for \(\gamma_{\mathbb{F}}(k)\), namely, NEWLINE\[NEWLINE\lceil(k+1)/2\rceil\;(\text{for }\mathbb{F}= \mathbb{C})\text{ or }k\;(\text{for }\mathbb{F}= \mathbb{R})\leq \gamma_{\mathbb{F}}(k)\leq 2k-1.NEWLINE\]NEWLINE Then it is shown that, for \(k=2\) and \(3\), the lower bounds for \(\gamma_{\mathbb{F}}(k)\) are attained. For a given set of \(k\) linearly independent selfadjoint operators \(X_1,\dots, X_k\) on \(H\), the set of all \(\gamma_{\mathbb{F}}(k)\)-dimensional subspaces \(H_0\) of \(H\) with \(X_j[H_0]\), \(1\leq j\leq k\), linearly independent is an open dense subset of the Grassmannian of all \(\gamma_{\mathbb{F}}(k)\)-dimensional subspaces of \(H\). Finally, the proved \(k= 3\) case is used to show that the ratio numerical range \(W(A/B)\equiv \{\langle Ax,x\rangle/\langle Bx,x\rangle: \langle Bx,x\rangle\neq 0\}\) of operators \(A\) and \(B\,(\neq 0)\) on \(H\) either has a nonempty interior or lies in a line or a circle.
scientific article; zbMATH DE number 6034492