Isomorphism of Groups of Operators on Hilbert Space

Prabhat Kumar; Amar Nath Kumar; Lalit Kumar Sharan

doi:https://doi.org/10.14445/22315373/IJMTT-V65I3P509

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 65 | Issue 3 | Year 2019 | Article Id. IJMTT-V65I3P509 | DOI : https://doi.org/10.14445/22315373/IJMTT-V65I3P509

Isomorphism of Groups of Operators on Hilbert Space

Prabhat Kumar, Amar Nath Kumar, Lalit Kumar Sharan

Citation :

Prabhat Kumar, Amar Nath Kumar, Lalit Kumar Sharan, "Isomorphism of Groups of Operators on Hilbert Space," International Journal of Mathematics Trends and Technology (IJMTT), vol. 65, no. 3, pp. 60-65, 2019. Crossref, https://doi.org/10.14445/22315373/IJMTT-V65I3P509

Abstract

In this paper we have established two theorems by using the role of identity operator analogous to that for identity element e of a group. Using the property of isomorphism efforts have also been made to establish a result according to that the order of an element of a group is equal to the order of the image of that group. It has also been established that the f image of an identity operator of a domain group is an identity element of co-domain group. The same kind of result is also established for the additive inverse of a group considered. It has also been observed a relation between homomorphism and abelian group. In fact this result is an analogous result imposing a stronger condition on the homomorphism we have established a set of necessary and sufficient conditions for the group to be an abelian. Efforts have also been made by establishing a result that the inverse of an isomorphism is again an isomorphism. We have also observed by establishing a result that the product of two isomorphism is isomorphism.

Keywords

Linear Space, Linear Transformation, Operator, Non singular Transformation, Norm, Normed Linear Space, Cauchy Sequence, Complete Metric Space, Banach Space, Inner Product, Hilbert Space, Continuous Linear Transformation, Operator on a Hilbert Space, Group. Homomorphism, Isomorphism.

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