The pseudo-spectrum of operator pencils

In this paper, we propose a new definition of the pseudo-spectrum for operator pencils, which is associated with two bounded operators [Formula: see text] and [Formula: see text] defined in a Hilbert space. Unlike other definitions available in the literature, we prove, under specific conditions on [Formula: see text] and [Formula: see text], that the pseudo-spectrum for operator pencils is equal to an [Formula: see text]-neighborhood of the generalized spectrum. Moreover, we demonstrate how to use this concept to redefine the pseudo-spectrum of an unbounded operator. We illustrate its usefulness through a numerical example dealing with the Schrödinger operator.