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

 International Journal of Mathematics Trends and Technology (IJMTT) © 2019 by IJMTT Journal Volume-65 Issue-12 Year of Publication : 2019 Authors : Isaiah Nalianya Sitati 10.14445/22315373/IJMTT-V65I12P519

MLA Style:Isaiah Nalianya Sitati  "Remarks and Specific Examples on Symmetric and Skew-Symmetric Operators in Hilbert Spaces" International Journal of Mathematics Trends and Technology 65.12 (2019):171-179.

APA Style: Isaiah Nalianya Sitati(2019). Remarks and Specific Examples on Symmetric and Skew-Symmetric Operators in Hilbert Spaces International Journal of Mathematics Trends and Technology, 171-179.

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.

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