Invertible perturbations of essential spectra for upper triangular relation matrices

This paper is concerned with the invertible perturbations of the essential spectra for the upper triangular partial relation matrix with given relations. A necessary and sufficient condition is given under which such a relation matrix admits a Fredholm (left Fredholm, or right Fredholm) relation completion by choosing some bounded invertible relation. It is shown that the essential (left essential, or right essential) spectrum under bounded perturbations can be a proper subset of that under invertible perturbations.