IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org ISSN (e): 2250-3021, ISSN (p): 2278-8719 Vol. 10, Issue 6, June 2020, ||Series -II|| PP 07-13 International organization of Scientific Research 7 | Page Intuitionistic Fuzzy Hyponormal Operator in IFH-Space A.Radharamani 1 , S. Maheswari 2 1 (Department of Mathematics, Chikkanna Govt. Arts College, India. 2 (Department of Mathematics, Tiruppur Kumaran College for Women, India. Received 10 June 2020; Accepted 27 June 2020 Abstract: In this work,we introduced the definition of Intuitionistic Fuzzy Hyponormal operator acting on an IFH-space, i.e.an operator  ∈ (ℋ) is Intuitionistic Fuzzy Hyponormalif  ∗  ≤ , ∀ ∈ ℋor equivalently  ∗  −  ∗ ≥ 0and given some elementary properties of Intuitionistic Fuzzy Hyponormal operator on an IFH-space. Also, we introduced some definitions like intuitionistic fuzzy invariant, eigenvalues, eigenvectors and eigenspaces which are related to Intuitionistic Fuzzy Hyponormal operator in IFH-space. Keywords:Intuitionistic Fuzzy Adjoint operator (IFA-operator), Intuitionistic Fuzzy Hilbert space (IFH- space), Intuitionistic Fuzzy Hyponormal operator (IFHN-operator),Intuitionistic FuzzyInvariant (IF-invariant), Intuitionistic Fuzzy Normal operator (IFN-operator), IntuitionisticFuzzy Self-Adjoint operator (IFSA-operator). I. INTRODUCTION In 1986, Atanossov [11] introduced the notion of intuitionistic fuzzy set. Park [10] introduced the notion of intuitionistic fuzzy metric space (, M, N, *, ⋄) with the use of continuous t-norm * and continuous t- conorm ⋄ in 2004. Saadati and Park [17] introduced modulation of the intuitionistic fuzzy metric space in IFH- space using continuous t representable in 2005. The new idea of intuitionistic fuzzy normed spaces was introduced by Goudarzi et al. [13] and introduced the modified definition of intuitionistic fuzzy inner product space (IFIP-space) with the help of continuous t-representable () in 2009. A triplet (ℋ, ℱ  , , ) where ℋis a real vector space,  is a continuous t -representable and ℱ  , is an Intuitionistic Fuzzy set on ℋ 2 × ℝ which was introduced by Goudarzi et al. [13] in 2009,and also Majumdar and Samanta [15] gave the various definition of IFIP-space and some of their properties using(ℋ, ,  ∗ ). The definition of IFH-space first introduced by Radharamani et al. [1] in 2018, and also some properties of IFA & IFSA operators in IFH-space by Radharamani et al.[2]. Then Radharamani et al. [3]introduced the concept of Intuitionistic Fuzzy Normal operator in 2020.An operator ∈ (ℋ) if it commutes with its Intuitionistic fuzzy adjoint operator.i.e,  ∗ =  ∗  and their properties. In 2020,Radharamani et al. [4],[5]giventhe definition of Intuitionistic Fuzzy Unitary operator (IFU-operator) and Intuitionistic Fuzzy Partial Isometry (IFPI-operator) on IFH-space ℋ, and gave some properties of these operators in IFH-space and also the relation with isometric isomorphism of ℋ on to itself. In this paper,we consider an Intuitionistic fuzzy normal operator in IFH-space and introduced the definition of Intuitionistic Fuzzy hyponormal operator(IFHN- operator) and we provided some important properties of IFHN- operator on IFH-space. And also introduce intuitionistic fuzzy invariant and eigenvectors and eigenspaces which is using in Intuitionistic Fuzzy Hyponormal Operator in IFH-space, which all are discussed in detail. The classification of this paper is as follows: Section 2 provides some preliminary definitions and theorems which are used in this paper. In section 3, we introduced the concept of Intuitionistic Fuzzy hyponormal operator(IFHN- operator) and prove some properties of Intuitionistic fuzzy hyponormal operator have been studied. II. PRELIMINARIES Definition 2.1: [13] IFIP-space Let : ℋ 2 × 0, +∞ → 0,1 and : ℋ 2 × (0, +∞) → [0,1] be Fuzzy sets, such that , ,  + , ,  ≤ 1, ∀ , ∈ℋ & >0. An Intuitionistic Fuzzy Inner Product Space (IFIP-Space) is a triplet (ℋ, ℱ  , , ), where ℋ is a real vector space,  is a continuous t -representable and ℱ  , is an Intuitionistic Fuzzy set on ℋ 2 × ℝ satisfying the following conditions for all u, , ∈ℋ and s, ,  ∈ ℝ: (IFI – 1) ℱ  , , ,0 =0 and ℱ  , , ,  >0, for every  >0. (IFI - 2) ℱ  , , ,  = ℱ  , , , . (IFI - 3) ℱ  , , ,  ≠ H for some ∈ℝ iff u≠ 0, where H =  1, if  >0 0, if ≤ 0