Weyl's Theorem for Bounded Self-Adjoint Operators

Let $T$ be a bounded self-adjoint operator on a separable infinite-dimensional right quaternionic Hilbert space, and let $\text{sigdS}(T)$ be the set of right eigenvalues of finite type. We prove that Weyl’s theorem holds for $T$: $$\text{sigeS}(T)=\text{sigS}(T)\setminus \text{sigdS}(T),$$ which completes the inclusion obtained in the general case. The proof proceeds through a single three-way criterion: a point of $\text{sigS}(T)$ lies outside $\text{sigeS}(T)$, and equally lies in $\text{sigdS}(T)$, exactly when it is isolated with a spectral atom of finite rank. The natural route through Riesz projections is not available as it stands, because a Riesz projection need not be orthogonal; we show that for a self-adjoint operator it in fact is, the Riesz projection of an isolated spectral part coinciding with the corresponding spectral projection, so that the algebraic and geometric multiplicities agree. A $2\times 2$ quaternionic matrix shows that this last identity fails without self-adjointness, and a normal operator with genuinely spherical $S$-spectrum shows what has to change in the normal case.