Journal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218 Volume 12, Issue 4 available at www.scitecresearch.com/journals/index.php/jprm 2030| SCITECH Volume 12, Issue 4 RESEARCH ORGANISATION Published online: November 03, 2017| Journal of Progressive Research in Mathematics www.scitecresearch.com/journals METRIC EQUIVALENCE AS AN ALMOST SIMILARITY PROPERTY Eric M.Gitonga 1 , Sammy W. Musundi 1* , Benerd M. Nzimbi 2 1 Department of Physical Sciences, Chuka University, P.O. Box 109-60400, Kenya. Email:gitongaeric420@gmail.com. 1* Department of Physical Sciences, ChukaUniversity, P.O. Box 109-60400, Kenya. Email: sammusundi@yahoo.com. 2 School of Mathematics, College of Biological and Physical Sciences, University of Nairobi, P. O. Box 30197-00100,Nairobi. Email: nzimbi@uonbi.ac.ke. ABSTRACT Various results that relate to almost similarity and other classes of operators such as isometry, normal, unitary and compact operators have been extensively discussed. It has been shown that if operators S and T are unitarily equivalent, then S is almost similar to T. Similarly, it has been shown that if operators A and B are such that A is almost similar to B and if A is Hermitian, then A and B are said to be unitarily equivalent. Metric equivalence property which is a new relation in operator theory has drawn much attention from mathematicians in the recent past. Two operators S and T are unitarily equivalent if they are metrically equivalent projections. It has been shown that if operators S and T are unitarily equivalent, then S is metrically equivalent to T. However, there is no literature that has been shown for the conditions under which metric equivalence and almost similarity coincide. In this paper we will therefore strive to establish the equivalence relation between metric equivalence property and almost similarity relation. To achieve this, properties of invertible operators, normal operators, similar operators, unitarily operators as well as projection and self- adjoint operators will be employed. Mathematics Subject Classification: 47A05, 47B15, 47B25. Keywords: Almost similarity relation; Unitarily equivalent relation; Metric equivalence property. 1. INTRODUCTION The class of almost similar operators was first introduced by (Jibril, 1996). He defined the class of almost similar operators as follows: