Norm Estimates for Solutions of Polynomial Operator Equations

We consider the equations∑k=0mcm-kAkXBk=Cand∑k=0mcm-kAkXBm-k=C, whereck∈C (k=1,…,m),c0=1,A,B,Care given linear bounded operators in a Banach space andXis to be found. Representations of solutions are derived. In the cases of Euclidean and Hilbert spaces, norm estimates for the solutions are suggested.

Estimates for lower bounds of eigenvalues of the poly-Laplacian and the quadratic polynomial operator of the Laplacian

In this paper, we investigate the Dirichlet eigenvalue problems of the poly-Laplacian with any order and the quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first k eigenvalues.