On the ideals of strictly singular and inessential operators

In the present paper, we prove that if $X$ is a subprojective Banach space, then the ideal of strictly singular operators on $X$ is equal to the ideal of inessential operators on $X$. We give an example to show that equality does not hold for all Banach spaces $X$. We also investigate the relationship between the semi-Fredholm operators on a Banach space and the right and left null divisors in the quotient algebra of all the bounded operators modulo the ideal of compact operators. We are able to get some complete characterizations of the null divisors when the Banach space is subprojective.