Integral and differential test for convergence of operator series (Q2443490)

The author presents some operator analogs of the classical series convergence tests for numerical series. The results are for series with operator terms \(\sum _{n=1}^{\infty }A_{n} \), where \(A_{n} \) are bounded (and in some cases positive) operators defined on a real Banach space, partially ordered by a positive cone. In particular, the author obtains an operator analog of the integral test for series of the form \(\sum _{n=1}^{\infty }F(n) \), where \(F(t)\) is an operator-valued function on \([1, \infty )\).