New Convergence Mode For Generalized Spectrum Approximation

Abstract

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 the old methods, those required in this paper are weaker. In addition, we prove that the property \(U\) holds, which solves a spectral pollution problem arising in the spectrum approximation of unbounded operators.