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

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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

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.

References

[1] ALBERT,A.A (i) Modern higher Algebra (cambridge), 1938.
[2] BHAMBRI, S.K. (i) A course in abstract algebra, vikas publishing House, Pvt.LTD.
[3] BIKHOFF AND MACLANE (i) A survey of modern Algebra, Fourth edition, Macmillen publishing co.
[4] CHANDRA,P. (i) Investigation in tyo the theory of operators and linear spaces, Ph.D. Thesis, Patna University (1977).
[5] CHEVALLEY,C. (i) Fundamental concepts of algebra Academic press inc.,New york (1956).
[6] CLARKSON, J.A (i) uniformly convex spaces, Translation, Americal Math soc., 40(1936),396-414.
[7] COOKE, R.G. (i) linear operaters, Macmillan, london,1953.
[8] COOPER, J.L.B. (i) The spectral Analysis of self -adjoint operators, Q.J.M.(oxford), 16, (1945), 31-48 (ii) symmetric operators in Hilbert space, Proceedings of the London Mathematical society., (2), 50,(1948),11-55.
[9] DAY, M.M. (i) Operations in Banach Spaces, Transaction of the American Mathematical society.,51, (1942) 583-608.
[10] DUNFORD,N AND J.T.SCHWARTZ. (i) Linear operators, Vol.1 General theory, Interscience, New York, 1958.
[11] EDWARD, R.E. (i) Functional analysis Holt Richart and Winston, Inc., New York 1965.
[12] HALMOS, P.R. (i) Finite dimensional vector space, Van Nostrand, Princeton, N.J., (1958).
(ii) A Hilbert space, Problems Book, springer international student edition, Narosa Publishing House New Delhi 1978.
[13] JHA,K.K.(i) Advanced course in modern Algebra (New Bharti Prakashan patna, Delhi (2002).
(ii) Functional Analysis (students Friends, Patna,(1986).
[14] RUDIN, W. (i) Functional Analysis, Tata Mc.Graw Hill Publishing company Ltd., New Delhi (1974).
[15] SIMMONS, G. F. (i) Introduction to Topology and Modern Analysis. (Mc Graw-Hill Book company, Inc., New York, (1963).

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