New convergence mode for the generalized spectrum approximation

In this paper, we introduce a new convergence mode to deal with the generalized spectrum approximation of two bounded operators. This new technique is obtained by extending the well-known $\nu$-convergence used in the case of classical spectrum approximation. This new vision allows us to see the $\nu$-convergence assumption as a special case of our new method compared to the hypotheses needed in old methods, those required in this paper are weaker. In addition, we prove that the property $U$ holds, which solves the spectral pollution problem arising in spectrum approximation of unbounded operator.