On the spectral $\nu$-continuity

Abstract

The spectrum of elements of Banach algebras can be considered as an application that has as codomain the space of compact subsets of $\mathbb{C}$ with the Hausdorff metric, the essential question to answer, which opens this line of research, is whether this mapping is continuous, on this line, this paper studies whether some subsets of the spectrum are $\nu$-continuous at certain operators, conceptualizing:  the spectrum $\sigma$ is called $\nu$-continuous if a sequence $(T_n)$ is $\nu$-convergent to $T$ implies $\sigma(T_n)\to \sigma(T)$ in the Hausdorff metric; however, the results of the present paper demonstrate that special care is required about the zero point in the spectrum to guarantee this convergence, which is why additional conditions are requisite on this singular point; for example, it is known that the spectrum is upper semi-$\nu$-continuous, this paper shows that the spectrum is also lower semi-$\nu$-continuous (hence $\nu$-continuous) at a Fredholm operator for which 0 is an accumulation point of the spectrum, and which satisfies a condition on the spectrum similar to one imposed by Conway and Morrey; in addition, in this manuscript, it is established that the approximate point spectrum $\sigma_{ap}$ is upper semi-$\nu$-continuous except possibly for the zero point in the spectrum and shows the conditions on a Fredholm operator to ensure the approximate point spectrum $\sigma_{ap}$ is $\nu$-continuous. Finally, it is shown that the lower semi-$\nu$-continuity of the Weyl spectrum can be obtained by restricting to essentially $G_1$ operators and if a sequence of $p$-hyponormal operators $T_n$ (which are uniformly bounded below on the complement of the kernel) $\nu$-converges to a Fredholm operator $T$ (for which $0\in \sigma_{ap}(T)$), then  $\sigma(T_n)$ converges to $\sigma(T)$.