This is an announcement for the paper "On the divergence of series of the form sum_{k=1}^infty||A_k x||^p" by Ivan Feshchenko. Abstract: Let {A} be a system of operators. With any element x we associate the set of elements {Ax}. We study conditions under which there exists an element x such that the sum of p-th powers of norms of the elements {Ax} is equal to infinity. Archive classification: math.FA Mathematics Subject Classification: 40H05, 46B20, 47A05 Remarks: 9 pages, submitted to Studia Mathematica Submitted from: ivanmath007@gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.1863 or http://arXiv.org/abs/1208.1863