Eigenvectors of perturbed operators (Q1290962)

The main result of this article constitutes the following theorem: Let \(A:H\rightarrow H\) be a compact selfadjoint operator and \(f\) a strictly increasing differentiable function in the interval \(I.\) For \(z\in H\) form a family of compact selfadjoint operators \(A(t)=A+f(t)\langle\cdot,z \rangle z\). If \(z\) is not an eigenvector of \(A\) for each \(t\in I,\) there exists the maximal eigenvalue \(\lambda_{\max} (t)\) and there does not exist a parameter \(t\) for which \(z\) is orthogonal to \(Y_{t}=\text{Ker} (A(t)-\lambda_{\max} (t))\), then we can find for each \(t\) an eigenvector \(x_{\max}(t)\in Y_{t}\), which is piecewise differentiable as the function of \(t\) and for which \(\langle x_{\max}(t),z\rangle\equiv 1.\) Moreover, \(\|x_{\max}(t)\|\) is a strictly decreasing function on \(I\).