Approximation numbers of an infinite matrix on HpB spaces

In this paper, we study conditions that an infinite matrix has to satisfy in order to define a linear bounded operator from any weighted HpB space[1] with a weighted sequence of positive numbers into itself, and give upper and lower estimations of the approximation numbers of such an infinite matrix. Studying approximation numbers of an infinite matrix operator is one of the important studies in Functional Analysis. In this paper, we evaluate the norm of such operator on different spaces by calculating upper and lower estimations for this norm on different spaces with different values of p, B then we give upper and lower estimations for the approximation numbers of this operator and its dual by using Hahn Banach theorem and useful lemmas concerning reordering of any sequence of real numbers and relating this recording with the norm also we make use of the properties of approximation numbers of any operator.