Trace formula for polynomial operator pencils and its applications (Q1180105)

Let \(H\) be separable Hilbert space, \(P_ j\) linear operators, \(P_ m=E\), \(\lambda_ 1,\lambda_ 2,\dots\) eigenvalues of the pencil \(L(\lambda)=\sum_{j=0}^ m \lambda^ j P_ j\). Under certain assumptions the formula \[ \sum_{j=1}^ \infty(\lambda_ j-z)^{-q- 1}=-{1\over q!}\text{sp} W_ q(z),\tag{1} \] where \[ W_ q(z)={d^ q\over dz^ q} [L'(z)L^{-1}(z)]\in\sigma_ 1(H) \] is proved. Applications of formula (1) are given.