Remarks and Specific Examples on Symmetric and Skew-Symmetric Operators in Hilbert Spaces

Isaiah Nalianya Sitati

doi:https://doi.org/10.14445/22315373/IJMTT-V65I12P519

International Journal of Mathematics Trends and Technology

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Volume 65 | Issue 12 | Year 2019 | Article Id. IJMTT-V65I12P519 | DOI : https://doi.org/10.14445/22315373/IJMTT-V65I12P519

Remarks and Specific Examples on Symmetric and Skew-Symmetric Operators in Hilbert Spaces

Isaiah Nalianya Sitati

Abstract

In this paper, classes of symmetric and skew-symmetric operators on a Hilbert Space are characterised. It is demonstrated that skew-symmetric operators admit skew-symmetric matrix representation with respect to some orthonormal basis. It will also be shown that the characteristic polynomial of a self adjoint operator on an n-dimensional Hilbert Space, H has n real zeros (counted with multiplicity).Further, a specific example of a normal form of a skew-adjoint operator shall be given and then be shown that the rank of a skewsymmetric operator is always even. By considering a forward shift operator on a Hilbert space, it is demonstrated that not every skew-symmetric operator is biquasitriangular.Finally, the relationship between complex symmetric and skew-symmetric operators is established.

Keywords

Self- adjoint operators, Hilbert space, Complex-symmetric operators, Skewsymmetric operators, Bitriangular and Conjugation operators.

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Citation :

Isaiah Nalianya Sitati, "Remarks and Specific Examples on Symmetric and Skew-Symmetric Operators in Hilbert Spaces," International Journal of Mathematics Trends and Technology (IJMTT), vol. 65, no. 12, pp. 171-179, 2019. Crossref, https://doi.org/10.14445/22315373/IJMTT-V65I12P519