Pseudo S-spectra of special operators in quaternionic Hilbert spaces.

For a bounded quaternionic operator T on a right quaternionic Hilbert space H and ε > 0 , the pseudo S -spectrum of T is defined as Λ ε S (T) : = σ S (T) ⋃ { q ∈ H ∖ σ S (T) : ‖ Q q (T) − 1 ‖ ≥ 1 ε } , where H denotes the division ring of quaternions, σ S (T) is the S -spectrum of T and Q q (T) = T 2 − 2 Re (q) T + | q | 2 I. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate several properties of the pseudo S -spectrum and explicitly compute the pseudo S -spectra for some special classes of operators such as upper triangular matrices, self adjoint-operators, normal operators and orthogonal projections. In particular, by an application of S -functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number r if and only if for every ε > 0 the pseudo S -spectrum of the operator is the circularization of a closed disk in the complex plane centred at r with the radius ε. Further, we propose a G 1 -condition for quaternionic operators and prove some results in this setting. [ABSTRACT FROM AUTHOR]