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The paper aims to extend the concept of Fredholm, Weyl and Jeribi essential spectra in the quaternionic setting. Furthermore, some properties and stability of the corresponding spectra of Fredholm and Weyl operators have been investigated in this setting. To achieve the goal, a characterization of the sum of two invariant bounded linear operators has been obtained in order to explore various properties of the Fredholm operator and Weyl operator under some assumptions in quaternionic setting. Also, various sequential properties of the pseudo-resolvent operator, right quaternionic linear operator, Weyl operator, Weyl S-spectrum, Jeribi essential S-spectrum and some properties of [Formula: see text] block operator matrices have been discussed. The spectral mapping theorem of essential S-spectrum, Weyl S-spectrum and Jeribi essential S-spectrum for self-adjoint operators has been established. A characterization of the essential S-spectrum and Weyl S-spectrum of the sum of two bounded linear operators concludes this investigation.

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In this paper, we show that the spectral mapping theorem holds for ( m , n ) * {(m,n)^{*}} -paranormal operators. We also exhibit the self-adjointness of the Riesz idempotent E λ {E_{\lambda}} of ( m , n ) * {(m,n)^{*}} -paranormal operators concerning for each isolated point λ of σ ⁢ ( T ) {\sigma(T)} . Moreover, we show Weyl’s theorem for ( m , n ) * {(m,n)^{*}} -paranormal operators and f ⁢ ( T ) {f(T)} for every f ∈ ℋ ⁢ ( σ ⁢ ( T ) ) {f\in\mathcal{H}(\sigma(T))} . Furthermore, we investigate the class of totally ( m , n ) * {(m,n)^{*}} -paranormal operators and show that Weyl’s theorem holds for operators in this class.

In the present paper, we prove spectral mapping theorem for (m,n)-paranormal operator T on a separable Hilbert space, that is, f (?w(T)) = ?w(f(T)) when f is an analytic function on some open neighborhood of ?(T). We also show that for (m,n)-paranormal operator T, Weyl?s theorem holds, that is, ?(T)-?w(T) = ?00(T). Moreover, if T is algebraically (m,n)-paranormal, then spectral mapping theorem and Weyl?s theorem hold.