ss-numbers of shift operators of formal entire functions

The main objective of this work is to find upper bounds for ss-numbers of infinite series of the forward shift operator MzMz over the space Hβp, 1≤p<∞1≤p<∞, of formal power series f(z)=∑k=0∞akzkβ(k), equipped with the norm ‖f‖=(∑k=0∞|ak|p)1p<∞, where {β(k)}{β(k)} is a sequence of positive numbers such that β(0)=1β(0)=1. This is done by giving exact estimations of ss-numbers of powers of MzMz. We apply our results to get upper estimations of ss-numbers of some examples of entire functions such as the sine and exponential functions considered as a type of a right shift operator.