Pseudo S-spectra of special operators in quaternionic Hilbert spaces

Abstract

For a bounded quaternionic operator T on a right quaternionic Hilbert space H and ε>0, the pseudo S-spectrum of T is defined asΛεS(T):=σS(T)⋃{q∈H∖σS(T):‖Qq(T)−1‖≥1ε}, where H denotes the division ring of quaternions, σS(T) is the S-spectrum of T and Qq(T)=T2−2Re(q)T+|q|2I. 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 G1-condition for quaternionic operators and prove some results in this setting.