On the pseudo-similarity of bounded linear operators

In order to extend the notion of similarity to a more general concept, we introduce a new notion called the pseudo-similarity. We consider various characterizations and representations of pseudo-similar operators. Knowing that the similarity preserves the spectrum, we establish in the case of pseudo-similarity a more general result, giving certain relationships between the spectrum of pseudo-similar operators. Furthermore, we delve into the pseudo-similarity to some particular operators. Particularly, we focus on some upper triangular operator matrices and we determine their spectrum as an application of our findings. Finally, further applications to the spectrum of pseudo-similar operators are established.