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 aswhere H denotes the division ring of quaternions, σ S (T ) is the S-spectrum of T and ∆ q (T ) = T 2 − 2Re(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 Sspectrum of the operator is the circularization of a closed disc in the complex plane centered at r with the radius √ ε. Further, we propose a G 1 -condition for quaternionic operators and prove some results in this setting.