### A Convergence Theorem for Selfadjoint Operators Applicable to Dirac Operators with Cutoff Potentials.

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From Corollary 3.5 in [Berkani, M; Sarih, M.; Studia Math. 148 (2001), 251–257] we know that if $S$, $T$ are commuting $B$-Fredholm operators acting on a Banach space $X$, then $ST$ is a $B$-Fredholm operator. In this note we show that in general we do not have $error\left(ST\right)=error\left(S\right)+error\left(T\right)$, contrarily to what has been announced in Theorem 3.2 in [Berkani, M; Proc. Amer. Math. Soc. 130 (2002), 1717–1723]. However, if there exist $U,V\in L\left(X\right)$ such that $S$, $T$, $U$, $V$ are commuting and $US+VT=I$, then $error\left(ST\right)=error\left(S\right)+error\left(T\right)$, where $error$ stands for the index of a $B$-Fredholm operator.

A characterization of compactness of a given self-adjoint bounded operator A on a separable infinite-dimensional Hilbert space is established in terms of the spectrum of perturbations. An example is presented to show that without separability, the perturbation condition, which is always necessary, is not sufficient. For non-separable spaces, another condition on the self-adjoint operator A, which is necessary and sufficient for the perturbation, is given.

An approach to a local analysis of solutions of a perturbation problem is proposed when the unperturbed operator has affine symmetries. The main result is a local theorem on existence, uniqueness, and analytic dependence on a parameter.

We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties $\left(\mathrm{SBaw}\right)$, $\left(\mathrm{SBab}\right)$, $\left(\mathrm{SBw}\right)$ and $\left(\mathrm{SBb}\right)$ are not preserved under direct sums of operators. However, we prove that if $S$ and $T$ are bounded linear operators acting on Banach spaces and having the property $\left(\mathrm{SBab}\right)$, then $S\oplus T$ has the property $\left(\mathrm{SBab}\right)$ if and only if ${\sigma}_{{\mathrm{SBF}}_{+}^{-}}(S\oplus T)={\sigma}_{{\mathrm{SBF}}_{+}^{-}}\left(S\right)\cup {\sigma}_{{\mathrm{SBF}}_{+}^{-}}\left(T\right)$, where ${\sigma}_{{\mathrm{SBF}}_{+}^{-}}\left(T\right)$ is the upper semi-B-Weyl spectrum of $T$. We obtain analogous preservation results for the properties $\left(\mathrm{SBaw}\right)$, $\left(\mathrm{SBb}\right)$ and $\left(\mathrm{SBw}\right)$ with...