Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 129 https://internationalpubls.com Fuzzy Soft Paranormal Operator in Fuzzy Soft Hilbert Space Dr A Radharamani1, T Nagajothi2 1Assistant Professor Department of Mathematics Chikkanna Government Arts College, Tirupur – 641602 Mail id: radhabtk@gmail.com 2Assistant Professor Department of Mathematics PSG College of Arts & Science, Coimbatore – 641014 Mail id: cnpp1977@gmail.com Article History: Received: 23-01-2024 Revised: 08-04-2024 Accepted: 26-04-2024 Abstract: This paper defines the fuzzy soft paranormal operator and discusses several fundamental fuzzy soft paranormal operator properties in fuzzy soft hilbert space. Some concepts relevant to the fuzzy soft paranormal operator have been defined in fuzzy soft Hilbert space. Keywords: Fuzzy soft normal operator, fuzzy soft Hilbert space, fuzzy soft hyponormal operator, fuzzy soft paranormal operator I INTRODUCTION More than a century ago, the field of functional analysis was established to address a number of problems in pure mathematics. In addition to regularly presenting us with uncertainty, the phenomena under study's ambiguity also provides us with instruments for assessing faults in solutions to issues with both infinite and limited dimensions. In a variety of fields, including engineering, business, medicine, and economics, this kind of problem might be encountered. Our conventional mathematical methods frequently fall short in addressing such problems. Thus, L. Zadeh[3] provided an extension of set theory in 1965. Fuzzy set theory was the term given to the resulting theory. Fuzzy set theory quickly established itself as an effective method for dealing with ambiguous circumstances. The basis function from a set x to a set [0,1] defines the set x in classical set theory. In contrast, a set in fuzzy set theory is described by its membership function, which ranges from x to the closed range between 0 and 1. In 1999, Molodtsov[4] also developed a fresh generalisation for dealing with uncertainty. Soft set theory was created as a result of this research. Since then, it has been applied to tackle difficult issues in a number of fields, including computer science, engineering, medicine, and others. A soft set is a collection of universal sets that has been parametrized. Soft set gave rise to the ideas of soft point, soft normed space, soft inner product space, and soft Hilbert space, which were later applied in functional analysis to tackle a number of different mathematical topics. The concept of a fuzzy soft set was initially introduced in 2001 by Maji[5] et al. The idea was created by using a soft set and a fuzzy set. To provide more precise and thorough findings, it was necessary to merge the two concepts. Fuzzy soft point[6] and fuzzy soft normed space[7] were created as a result of the framework's expansion to include these new concepts. Faried Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 130 https://internationalpubls.com [10]et al. presented fuzzy soft Hilbert spaces in 2020. The fuzzy soft linear operators are also included. We introduce a brand-new class of fuzzy soft paranormal operator and establish a number of associated theorems in this article. II PRELIMINARIES This section serves as a preface to the topic that follows by providing specific notations, definitions, and preliminaries for fuzzy set, soft set, and fuzzy soft set. Definition 2.1: [3] Fuzzy set Let Ԏ be a universal set. A fuzzy set ₳ ̌ over Ԏ is a set characterized by a function 𝜂₳ ̌: Ԏ → [0,1]. 𝜂₳ ̌ is called the membership, characteristic or indicator function of the fuzzy set ₳ ̌ and the value 𝜂₳ ̌(𝔵) is termed the grade of membership of 𝔵 ∈ Ԏ in ₳ ̌. Definition 2.2: [4, 10] Soft set Assume that 𝒫(Ԏ ) the power set of Ԏ and E be the collection of parameters and ⊆ 𝐸. The mapping ɠ: ₳ ̌ → բ(Ԏ ), where (ɠ, ₳ ̌) = {ɠ(𝑙) 𝜖 բ(Ԏ ): 𝑙 ∈ ₳ ̌}. As a result (ɠ, ₳ ̌) is called the soft set. Definition 2.3: [5] Fuzzy soft set LetԎ be a universal set, E be a set of parameters and ₳ ̌ ⊆ E. A pair (ɠ, ₳ ) is called a fuzzy soft set over Ԏ , where ɠ is a mapping given by ɠ: ₳ → ℱ(Ԏ), ℱ(Ԏ) is the family of all fuzzy subsets of Ԏ and the fuzzy subset of Ԏ is defined as a map 𝜂 from Ԏ to [0,1]. The family of all fuzzy soft sets (ɠ, ₳ ) over a universal set Ԏ , in which all the parameter sets ₳ ̌ are the same, is denoted by 𝐹𝑆𝑆(Ԏ )₳ ̌ = 𝐹𝑆𝑆(Ԏ ) Definition 2.4: [9] Fuzzy soft Hilbert space A fuzzy soft inner product space is defined as(Ԏ ̃ , 〈. , . 〉̃). This space, which is fuzzy soft complete in the induced fuzzy soft normed space called as a fuzzy soft Hilbert space and denoted by (�̃� , 〈. , . 〉̃). Every fuzzy soft Hilbert space is obviously a fuzzy soft Banach space. Definition 2.5: [2] Fuzzy soft linear operator in �̃� Consider �̃� to be a fuzzy soft Hilbert space. A fuzzy soft linear operator ₮̃: �̃� → �̃� is called a fuzzy soft linear operator in �̃� , then ₮̃ is a fuzzy soft linear operator on �̃� which is denoted as ₮̃ ∈̃ �̃�(�̃� ). ₮̃ is fuzzy soft bounded if there exists �̃� ∈̃ ℜ(₳ ) : ‖₮̃ (𝑙𝜂ɠ(𝑒) ) ̃ ‖ ≤̃ �̃� ‖𝑙𝜂ɠ(𝑒) ‖ ∀ 𝑙𝜂ɠ(𝑒) ∈̃ �̃� , then ₮̃ ∈̃ �̃�(�̃�) Definition 2.6: [2] Fuzzy soft adjoint operator in �̃� The fuzzy soft adjoint operator ₮̃∗̃of a fuzzy soft linear operator ₮̃ is defined by 〈₮̃𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(2) ̃ 〉 =̃ 〈𝑙1 𝜂1ɠ(𝑒1) , ₮̃∗̃𝑙2 𝜂2ɠ(2) ̃ 〉 for all 𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) ∈̃ �̃� Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 131 https://internationalpubls.com Definition 2.7:[11] Fuzzy soft Normal Operator Let �̃� be an FS Hilbert space and ₮̃ ∈̃ �̃�(�̃�) . Then, ₮̃ is said to be an FS normal operator if ₮̃₮̃∗̃ =̃ ₮̃∗̃₮̃ Definition 2.8: [11] Fuzzy soft self adjoint operator The FS-operator ₮̃ of FSH-space �̃� is called fuzzy soft self adjoint (FS-self adjoint operator) if ₮̃ =̃ ₮̃∗̃ Definition 2.9: [14] Fuzzy soft isometry operator Let �̃� be an FS Hilbert space and ₮̃ ∈̃ �̃�(�̃� ) . Then, ₮̃ is said to be an FS isometry operator if 〈₮̃𝑙1 𝜂1ɠ(𝑒1) , ₮̃̃ 𝑙2 𝜂2ɠ(𝑒2) 〉 =̃ 〈𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(2) ̃ 〉 for all 𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) ∈̃ �̃� Definition 2.10: [13] Fuzzy soft projection operator Consider �̃� to be a fuzzy soft Hilbert space. A fuzzy soft linear operator ₮̃: �̃� → �̃� is called a fuzzy soft projection operator in �̃� if ₮̃2 =̃ ₮̃ ie, ₮ ̃is an idempotent. Definition 2.11: [15] Fuzzy soft hyponormal operator Consider �̃� to be a fuzzy soft Hilbert space. ₮̃ ∈ �̃�(�̃� )is called fuzzy soft hyponormal operator if ‖₮̃∗̃𝑙𝜂ɠ(𝑒) ‖ ≤ ‖₮̃𝑙𝜂ɠ(𝑒) ‖ for all 𝑙𝜂ɠ(𝑒) ∈ ̃�̃� or equivalently ₮̃∗̃₮̃ ≥ ₮̃₮̃∗̃ Definition 2.12: [16] M-Fuzzy soft hyponormal operator Let �̃� be an FS Hilbert space and let ₮̃ ∈ �̃�(�̃� ) is called M – fuzzy soft hyponormal operator if there exist a real number ℳ, such that ‖(₮̃−̃₴̃𝐼) ∗̃ 𝑙𝜂ɠ(𝑒) ̃ ‖ ≤̃ ℳ̃ ‖(₮̃−̃₴̃𝐼)𝑙𝜂ɠ(𝑒) ̃ ‖ for all 𝑙𝜂ɠ(𝑒) ∈̃ �̃� and for all ₴̃ ∈̃ ℂ̃(₳) III Main results The definition of the fuzzy soft paranormal operator in fuzzy soft Hilbert space is provided in this section. Definition 3.1: Fuzzy Soft Paranormal Operator (FSPN) Let �̃� be an FS Hilbert space and let Ʈ̃ ∈ �̃�(�̃� ) then Ʈ̃ is a FSPN operator if ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ̃ ‖ ‖𝑙𝜂ɠ(𝑒) ̃‖ ≥̃ ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 for all 𝑙𝜂ɠ(𝑒) ∈̃ �̃� Note: An operator Ʈ̃ ∈ �̃�(�̃� ) and �̃� be a FSHS then Ʈ̃ is said to be an FSPN operator if ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ̃ ‖, for every unit vector 𝑙𝜂ɠ(𝑒) in �̃�. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 132 https://internationalpubls.com Remark: Let Ʈ̃ ∈ �̃�(�̃� ), �̃� =̃ 𝑙2(�̃�) ie) 𝑙2(�̃�) =̃ {𝑙𝜂𝔾(𝑒) =̃ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ : ∑ |𝑙𝑖𝜂𝑖ɠ(𝑒𝑖) ̃ | 2 < ∞,∞ 𝑖=1 𝑙𝑖𝜂𝑖ɠ(𝑒𝑖) ∈ 𝒞𝑛(𝒜)̃ } for 𝑙𝜂ɠ(𝑒) ∈̃ 𝑙2(�̃�), defined ‖𝑙𝜂ɠ(𝑒) ‖ ̃ =̃ 〈𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉1 2⁄̃ =̃ (∑ |𝑙𝑖𝜂𝑖ɠ(𝑒𝑖) | 2 ∞ 𝑖=1 ) 1 2⁄̃ Let Ʈ̃: �̃� → �̃� defined by Ʈ̃ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ =̃ (𝜃, 𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ ∀ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ∈̃ ̃ 𝑙2(�̃�) a) To find Ʈ̃ is linear Take 𝑙𝜂ɠ(𝑒) =̃ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ �̃�𝛾ɠ(𝑎) =̃ (�̃�1 𝛾1ɠ(𝑎1) , 𝑚2 𝛾2ɠ(𝑎2) … ) ̃ ∈̃ 𝑙2(�̃�) Ʈ̃ (𝑙𝜂ɠ(𝑒) + �̃�𝛾ɠ(𝑎) ) ̃ =̃ Ʈ̃ (𝑙1 𝜂1ɠ(𝑒1) + �̃�1 𝛾1ɠ(𝑎1) , 𝑙2 𝜂2ɠ(𝑒2) + 𝑚2 𝛾2ɠ(𝑎2) , …. ) ̃ =̃ (𝜃, 𝑙1 𝜂1ɠ(𝑒1) + �̃�1 𝛾1ɠ(𝑎1) , 𝑙2 𝜂2ɠ(𝑒2) + 𝑚2 𝛾2ɠ(𝑎2) , … . ) ̃ =̃ (𝜃, 𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ +̃ (�̃�1 𝛾1ɠ(𝑎1) , 𝑚2 𝛾2ɠ(𝑎2) … ) ̃ Ʈ̃ (𝑙𝜂ɠ(𝑒) + �̃�𝛾ɠ(𝑎) ) ̃ =̃ Ʈ̃ (𝑙𝜂ɠ(𝑒) ) ̃ +̃ Ʈ̃ (�̃�𝛾ɠ(𝑎) ) ̃ Ʈ̃ (𝛼 𝑙𝜂ɠ(𝑒) ) ̃ =̃ (𝜃, 𝛼𝑙1 𝜂1ɠ(𝑒1) , 𝛼�̃�2 𝜂2ɠ(𝑒2) … ) ̃ =̃ �̃� (𝜃, 𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2𝑣(𝑒2) … ) ̃ =̃ �̃� Ʈ̃ ( 𝑙𝜂ɠ(𝑒) ) ̃ b) To find Ʈ̃ is finite Take (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ ∈̃ 𝑙2(�̃�) ‖Ʈ̃ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … )‖ ̃ 2 =̃ ‖(𝜃, 𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … )‖ 2̃ =̃ ∑ |𝑙𝑖𝜂𝑖ɠ(𝑒𝑖) ̃ | 2 ∞ 𝑖=1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 133 https://internationalpubls.com =̃ ‖𝑙𝜂ɠ(𝑒) ‖ ̃ 2 ie) ‖Ʈ̃ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … )‖ ̃ 2 =̃ ‖𝑙𝜂ɠ(𝑒) ‖ ̃ 2 ‖Ʈ̃𝑙𝜂ɠ(𝑒) ‖ ̃ 2 =̃ ‖𝑙𝜂ɠ(𝑒) ‖ ̃ 2 iff ‖Ʈ̃𝑙𝜂ɠ(𝑒) ‖ ̃ =̃ ‖𝑙𝜂ɠ(𝑒) ‖ ̃ which implies Ʈ̃ is finite Therefore, Ʈ̃ ∈ �̃�(�̃� ) c) To find Ʈ̃ is FSPN Take (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ ∈̃ 𝑙2(�̃�) ‖Ʈ̃ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … )‖ ̃ 2 =̃ ‖(𝜃, 𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … )‖ 2̃ =̃ ∑ |𝑙𝑖𝜂𝑖ɠ(𝑒𝑖) ̃ | 2 ∞ 𝑖=1 =̃ ‖ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … )‖ ̃ 2 ⇔ ‖Ʈ̃ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … )‖ =̃ ̃ ‖ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … )‖ ̃ d) Take (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ ∈̃ 𝑙2(�̃�) ‖Ʈ̃ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … )‖ =̃ ̃ ‖(𝜃, 𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ ‖ Let Ʈ̃2 (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ =̃ Ʈ̃ (Ʈ̃ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ ) =̃ Ʈ̃ (𝜃, 𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ Ʈ̃2 (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ =̃ (𝜃, 𝜃𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ ‖ Ʈ̃2 (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ ‖ =̃ ‖(𝜃, 𝜃𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ ‖ ‖ Ʈ̃2 (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ ‖ =̃ ∑ |𝑙𝑖𝜂𝑖ɠ(𝑒𝑖) | ̃ ∞ 𝑖=1 e) Taken any (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ ∈̃ 𝑙2(�̃�) Ʈ̃ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ =̃ (𝜃, 𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … ) ̃ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 134 https://internationalpubls.com ‖Ʈ̃ (𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … )‖ ̃ 2 =̃ ‖(𝜃, 𝑙1 𝜂1ɠ(𝑒1) , 𝑙2 𝜂2ɠ(𝑒2) … )‖ 2̃ =̃ ∑ |𝑙𝑖𝜂𝑖ɠ(𝑒𝑖) ̃ | 2 ∞ 𝑖=1 From d) and e), we get ‖Ʈ̃(�̃�𝜂ɠ(𝑒) )̃ ‖ 2 ≤̃ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ̃ ‖ Therefore, Ʈ̃ is FSPN operator Theorem 3.3: ‖Ʈ̃3𝑙𝜂ɠ(𝑒) ‖ ̃ ≥̃ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ̃ ‖ ‖Ʈ̃ 𝑙𝜂ɠ(𝑒) ‖ ̃ for every unit vector 𝑙𝜂ɠ(𝑒) in �̃� Proof: For every unit vector 𝑙𝜂ɠ(𝑒) ∈̃ �̃� Let ‖Ʈ̃3𝑙𝜂ɠ(𝑒) ‖ ̃ 2 =̃ 〈Ʈ̃3𝑙𝜂ɠ(𝑒) , Ʈ̃3𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈Ʈ̃ Ʈ̃2𝑙𝜂ɠ(𝑒) , Ʈ̃Ʈ̃2𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈 Ʈ̃∗̃Ʈ̃ Ʈ̃2𝑙𝜂ɠ(𝑒) , Ʈ̃2𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈 Ʈ̃2 Ʈ̃2𝑙𝜂ɠ(𝑒) , Ʈ̃2𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈 Ʈ̃4𝑙𝜂ɠ(𝑒) , Ʈ̃2𝑙𝜂ɠ(𝑒) 〉̃ ≤̃ ‖Ʈ̃4𝑙𝜂ɠ(𝑒) ‖ ̃ ‖ Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ̃ ‖Ʈ̃3𝑙𝜂ɠ(𝑒) ‖ ̃ 2 ≥̃ ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 4 ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 (Since Ʈ̃ is FSPN operator) ⇒ ‖Ʈ̃3𝑙𝜂ɠ(𝑒) ‖ ̃ ≥̃ ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ‖Ʈ̃ 𝑙𝜂ɠ(𝑒) ‖ ̃ Hence ‖Ʈ̃3𝑙𝜂ɠ(𝑒) ‖ ̃ ≥̃ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ̃ ‖ ‖Ʈ̃ 𝑙𝜂ɠ(𝑒) ‖ ̃ Theorem 3.4: Let �̃� be a FS Hilbert space and let Ʈ̃ ∈ �̃�(�̃� ) be a FSPN operator. Then ‖Ʈ̃𝑘+1𝑙𝜂ɠ(𝑒) ‖ ̃ 2 ≥̃ ‖Ʈ̃𝑘𝑙𝜂ɠ(𝑒) ‖ ̃ 2 ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ̃ ‖ for every positive integer 𝑘 ≥ 1 and for every unit vector 𝑙𝜂ɠ(𝑒) in �̃�. Proof: Let Ʈ̃ ∈ �̃�(�̃� ) be a FSPN operator By using the induction hypothesis, we will prove the theorem. For the case 𝑘 = 1, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 135 https://internationalpubls.com ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ̃ 2 ≥̃ ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ̃ ‖ Now suppose that ‖Ʈ̃𝑘+1𝑙𝜂ɠ(𝑒) ‖ ̃ 2 ≥̃ ‖Ʈ̃𝑘𝑙𝜂ɠ(𝑒) ‖ ̃ 2 ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ̃ ‖ is valid for k. Then 𝑘 = 𝑘 + 1 ‖Ʈ̃𝑘+2𝑙𝜂ɠ(𝑒) ‖ ̃ 2 =̃ 〈Ʈ̃𝑘+2𝑙𝜂ɠ(𝑒) , Ʈ̃𝑘+2𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈(Ʈ̃𝑘) ∗̃ Ʈ̃𝑘+2𝑙𝜂ɠ(𝑒) , Ʈ̃2𝑙𝜂ɠ(𝑒) 〉 ̃ =̃ 〈(Ʈ̃∗̃) 𝑘 Ʈ̃𝑘+2𝑙𝜂ɠ(𝑒) , Ʈ̃2𝑙𝜂ɠ(𝑒) 〉 ̃ =̃ 〈Ʈ̃2𝑘+2𝑙𝜂ɠ(𝑒) , Ʈ̃2𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈Ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒) , Ʈ̃2𝑙𝜂ɠ(𝑒) 〉̃ ≤̃ ‖Ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒) ‖ ̃ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ̃ Since ‖Ʈ2𝑙𝜂ɠ(𝑒) ̃ ‖ ≥̃ ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ‖𝑙𝜂ɠ(𝑒) ̃‖ ∀ 𝑙𝜂ɠ(𝑒) ∈̃ �̃� , ‖Ʈ̃𝑘+2𝑙𝜂ɠ(𝑒) ‖ ̃ 2 ≥̃ ‖Ʈ̃𝑘+1𝑙𝜂ɠ(𝑒) ‖ ̃ 2 ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ So 𝑘 = 𝑘 + 1 is valid and the proof is complete by the mathematical induction. Lemma 3.5: Let Ʈ̃ ∈ �̃�(�̃� ) be a FSPN operator. Then Ʈ̃𝑛 is also FSPN for every integer 𝑛 ≥ 1 Proof: It is sufficient to show that if Ʈ̃ and Ʈ̃𝑘 is a FSPN then Ʈ̃𝑘+1 is also FSPN For every unit vector 𝑙𝜂ɠ(𝑒) in �̃� Let ‖Ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒) ‖ ̃ 2 =̃ 〈Ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒) , Ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈(Ʈ̃2(𝑘+1)) ∗̃ Ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉 ̃ =̃ 〈(Ʈ̃∗̃) 2(𝑘+1) Ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉 ̃ =̃ 〈Ʈ̃4𝑘+4𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈Ʈ̃4(𝑘+1)𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉̃ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 136 https://internationalpubls.com ≤̃ ‖Ʈ̃4(𝑘+1)𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ‖ ̃ ≤̃ ‖Ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒) ‖ ̃ ‖Ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ ie) ‖Ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒) ‖ ̃ 2 ≥̃ ‖Ʈ̃𝑘+1𝑙𝜂ɠ(𝑒) ‖ ̃ 4 ‖𝑙𝜂ɠ(𝑒) ‖ implies that ‖Ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒) ̃ ‖ ≥̃ ‖Ʈ̃𝑘+1𝑙𝜂ɠ(𝑒) ‖ ̃ 2 By the above lemma, so Ʈ̃(𝑘+1) is also FSPN. Theorem 3.6: Let Ʈ̃ ∈ �̃�(�̃� ) is a self-adjoint fuzzy soft operator then Ʈ̃ is FSPN. Proof: For any 𝑙𝜂𝔾(𝑒) in �̃� with ‖𝑙𝜂ɠ(𝑒) ̃‖ =̃ 1, we know that Ʈ̃ is a self-adjoint fuzzy soft operator ie) Ʈ̃ =̃ Ʈ̃∗̃ Let ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 =̃ 〈Ʈ̃𝑙𝜂ɠ(𝑒) , Ʈ̃𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃∗̃Ʈ̃𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃ Ʈ̃𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃2𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 ≤̃ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ implies that ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ̃ So Ʈ̃ is FSPN. Theorem 3.7: Let Ʈ̃ ∈ �̃�(�̃� ) be FSPN and fuzzy soft self adjoint operator then Ʈ̃∗̃ is FSPN. Proof: For any 𝑙𝜂ɠ(𝑒) ∈̃ �̃�, ‖𝑙𝜂ɠ(𝑒) ̃‖ =̃ 1 Let ‖Ʈ̃∗̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 =̃ 〈Ʈ̃∗̃𝑙𝜂ɠ(𝑒) , Ʈ̃∗̃𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃Ʈ̃∗̃𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 137 https://internationalpubls.com =̃ 〈(Ʈ̃∗̃) 2 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 ≤̃ ‖(Ʈ̃∗̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ ‖Ʈ̃∗̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖(Ʈ̃∗̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ implies that ‖Ʈ̃∗̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖(Ʈ̃∗̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ ie) ‖(Ʈ̃∗̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ ≥̃ ‖Ʈ̃∗̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 Therefore, Ʈ̃ is FSPN. Theorem 3.8: Let ₷̃ and Ʈ̃ ∈̃ �̃� is a FSPN operator and fuzzy soft self adjoint operator. Then a) ₷̃ +̃ Ʈ̃ b) ₷̃ .Ʈ̃ are also as FSPN. Proof: For every unit vector 𝑙𝜂ɠ(𝑒) ∈̃ �̃� We know that ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ̃ ≥̃ ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ‖₷̃ 2 𝑙𝜂ɠ(𝑒) ‖ ̃ ≥̃ ‖₷̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 and ₷̃ =̃ ₷̃ ∗̃ , Ʈ̃ =̃ Ʈ̃∗̃ a) To prove that ₷̃ +̃ Ʈ̃ is a FSPN operator Let ‖(₷̃ +̃ Ʈ̃)𝑙𝜂ɠ(𝑒) ̃ ‖ 2 =̃ 〈(₷̃ +̃ Ʈ̃)𝑙𝜂ɠ(𝑒) , (₷̃ +̃ Ʈ̃)𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈(₷̃ +̃ Ʈ̃) ∗̃ (₷̃ +̃ Ʈ̃) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈(₷̃ ∗̃ +̃ Ʈ̃∗̃) (₷̃ +̃ Ʈ̃) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈(₷̃ +̃ Ʈ̃) (₷̃ +̃ Ʈ̃) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 ≤̃ ‖(₷̃ +̃ Ʈ̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ implies that ‖(₷̃ +̃ Ʈ̃)𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖(₷̃ +̃ Ʈ̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ Therefore, ₷̃ +̃ Ʈ̃ is a FSPN operator. b) To prove that ₷̃.Ʈ̃ is a FSPN operator Let ‖(₷̃. Ʈ̃)𝑙𝜂ɠ(𝑒) ̃ ‖ 2 =̃ 〈(₷̃. Ʈ̃)𝑙𝜂ɠ(𝑒) , (₷̃. Ʈ̃)𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈(₷̃. Ʈ̃) ∗̃ (₷̃. Ʈ̃) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 138 https://internationalpubls.com =̃ 〈(Ʈ̃∗̃ ₷̃ ∗̃ ) (₷̃. Ʈ̃) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈(Ʈ̃ ₷̃) (₷̃. Ʈ̃) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈(₷̃. Ʈ̃) (₷̃. Ʈ̃) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 ≤̃ ‖(₷̃. Ʈ̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ ‖(₷̃. Ʈ̃)𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖(₷̃. Ʈ̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ implies that ‖(₷̃. Ʈ̃)𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖(₷̃. Ʈ̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ Therefore, ₷̃.Ʈ̃ is a FSPN operator. Theorem 3.9: Let Ʈ̃ ∈ �̃�(�̃� ) is a FSN operator then Ʈ̃ is a FSPN operator Proof: For every unit vector 𝑙𝜂ɠ(𝑒) ∈̃ �̃� Let ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 =̃ 〈Ʈ̃𝑙𝜂ɠ(𝑒) , Ʈ̃𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃∗̃ Ʈ̃𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈(Ʈ̃Ʈ̃∗̃) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃2𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 ≤̃ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ ie) ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ implies that ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ̃ Therefore, Ʈ̃ is FSPN. Theorem 3.10: Let Ʈ̃ ∈ �̃�(�̃� ) is a FSPN operator and FSHN. Then ‖Ʈ̃ ‖ ≥̃ ‖Ʈ̃∗̃‖ is a FSPN operator. Proof: For every unit vector 𝑙𝜂ɠ(𝑒) ∈̃ �̃� Let ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 =̃ 〈Ʈ̃𝑙𝜂ɠ(𝑒) , Ʈ̃𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃∗̃ Ʈ̃𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 ≥̃ 〈(Ʈ̃Ʈ̃∗̃) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 ≥̃ 〈Ʈ̃∗̃ 𝑙𝜂ɠ(𝑒) , Ʈ̃∗̃ 𝑙𝜂ɠ(𝑒) ̃ 〉 Since ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ̃ ≥̃ ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 and Ʈ̃∗̃Ʈ̃ − Ʈ̃Ʈ̃∗̃ ≥̃ 0̃ ∀ 𝑙𝜂ɠ(𝑒) ∈̃ �̃� Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 139 https://internationalpubls.com ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≥̃ ‖Ʈ̃∗̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ⇒ ‖Ʈ̃ 𝑙𝜂ɠ(𝑒) ‖ ̃ ≥̃ ‖Ʈ̃∗̃ 𝑙𝜂ɠ(𝑒) ‖ ̃ implies that ‖Ʈ̃ ‖ ≥̃ ‖Ʈ̃∗̃‖ Theorem 3.11: Let Ʈ̃ ∈ �̃�(�̃� ) is a FSPN operator and ₷̃ is unitarily equivalent to Ʈ̃ then ₷̃ is a FSPN. Proof: For ₷̃ is unitarily equivalent to Ʈ̃, we have ₷̃ =̃ �̃�Ʈ̃�̃�∗̃ For some unitarily equivalent to ₷̃ 2 =̃ �̃�Ʈ̃2�̃�∗̃ ⇒ ‖₷̃ 2 𝑙𝜂𝔾(𝑒) ‖ ̃ =̃ ‖�̃�Ʈ̃2�̃�∗̃ 𝑙𝜂𝔾(𝑒) ‖ Let ‖₷̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 =̃ ‖(�̃�Ʈ̃�̃�∗̃)𝑙𝜂ɠ(𝑒) ̃‖ 2 〈₷̃𝑙𝜂ɠ(𝑒) , ₷̃𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈(�̃�Ʈ̃�̃�∗̃)𝑙𝜂ɠ(𝑒) , (�̃�Ʈ̃�̃�∗̃)𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈(Ʈ̃�̃�∗̃)𝑙𝜂ɠ(𝑒) , �̃�∗̃�̃� (Ʈ̃�̃�∗̃)𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈(Ʈ̃�̃�∗̃)𝑙𝜂ɠ(𝑒) , (Ʈ̃�̃�∗̃)𝑙𝜂ɠ(𝑒) 〉̃ [since �̃� is FS isometry] =̃ 〈(Ʈ̃�̃�∗̃) ∗̃ (Ʈ̃�̃�∗̃)𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉 ̃ =̃ 〈�̃� Ʈ̃∗̃(Ʈ̃�̃�∗̃)𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉̃ =̃ 〈�̃�Ʈ̃2�̃�∗̃ 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉̃ ≤̃ ‖�̃�Ʈ̃2�̃�∗̃ 𝑙𝜂ɠ(𝑒) ‖ ‖ 𝑙𝜂ɠ(𝑒) ‖ ‖₷̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖�̃�Ʈ̃2�̃�∗̃ 𝑙𝜂ɠ(𝑒) ‖ ‖ 𝑙𝜂ɠ(𝑒) ‖ implies that ‖₷̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖₷̃ 2 𝑙𝜂ɠ(𝑒) ‖ ‖ 𝑙𝜂ɠ(𝑒) ‖ Hence ₷̃ is a FSPN Theorem 3.12: Let Ʈ̃ ∈ �̃�(�̃� ) is an invertible and FSPN operator. Then Ʈ̃−1 is also a FSPN. Proof: For every unit vector 𝑙𝜂𝔾(𝑒) ∈̃ �̃� Let ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 =̃ 〈Ʈ̃𝑙𝜂ɠ(𝑒) , Ʈ̃𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃∗̃ Ʈ̃𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃ Ʈ̃∗̃ 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃2 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 ≤̃ ‖Ʈ̃2 𝑙𝜂ɠ(𝑒) ‖ ‖ 𝑙𝜂ɠ(𝑒) ‖ 𝑙𝜂ɠ(𝑒) is replaced by Ʈ̃−2 𝑙𝜂ɠ(𝑒) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 140 https://internationalpubls.com ‖Ʈ̃Ʈ̃−2 𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖Ʈ̃2 Ʈ̃−2 𝑙𝜂ɠ(𝑒) ‖ ‖ Ʈ̃−2 𝑙𝜂ɠ(𝑒) ‖ ‖Ʈ̃−1𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖ 𝑙𝜂ɠ(𝑒) ‖ ‖ Ʈ̃−2 𝑙𝜂ɠ(𝑒) ‖ ‖Ʈ̃−1𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖ Ʈ̃−2 𝑙𝜂ɠ(𝑒) ‖ ‖ 𝑙𝜂ɠ(𝑒) ‖ implies that ‖ Ʈ̃−2 𝑙𝜂ɠ(𝑒) ‖ ‖ 𝑙𝜂ɠ(𝑒) ‖ ≥̃ ‖Ʈ̃−1𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ie) ‖ (Ʈ̃−1) 2 𝑙𝜂ɠ(𝑒) ‖ ‖ 𝑙𝜂ɠ(𝑒) ‖ ≥̃ ‖Ʈ̃−1𝑙𝜂ɠ(𝑒) ̃ ‖ 2 Hence Ʈ̃−1 is also a FSPN Theorem 3.13: If Ʈ̃∗̃𝟐 Ʈ̃2 ≥̃ (Ʈ̃∗̃Ʈ̃) 2 , then Ʈ̃ is FSPN operator Proof: For every unit vector 𝑙𝜂ɠ(𝑒) ∈̃ �̃� Let Ʈ̃∗̃𝟐 Ʈ̃2 ≥̃ (Ʈ̃∗̃Ʈ̃) 2 Ʈ̃∗̃𝟐 Ʈ̃2 − (Ʈ̃∗̃Ʈ̃) 2 ≥̃ 0̃ 〈(Ʈ̃∗̃𝟐 Ʈ̃2 − (Ʈ̃∗̃Ʈ̃) 2 ) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉 ̃ ≥̃ 0̃ 〈(Ʈ̃∗̃𝟐 Ʈ̃2) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉 − 〈((Ʈ̃∗̃Ʈ̃) 2 ) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉 ̃ ≥̃ 0̃ 〈(Ʈ̃∗̃𝟐 Ʈ̃2) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉 ≥̃ 〈((Ʈ̃∗̃Ʈ̃) 2 ) 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉 ̃ 〈Ʈ̃2 𝑙𝜂ɠ(𝑒) , Ʈ̃2𝑙𝜂ɠ(𝑒) 〉 ≥̃ 〈Ʈ̃∗̃Ʈ̃𝑙𝜂ɠ(𝑒) , Ʈ̃∗̃Ʈ̃ 𝑙𝜂ɠ(𝑒) 〉̃ since ‖Ʈ̃∗̃Ʈ̃̃‖ =̃ ‖Ʈ̃‖ 2 ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ 2 ≥̃ ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 4 ⇒ ‖Ʈ̃2𝑙𝜂ɠ(𝑒) ‖ ≥̃ ‖Ʈ̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 Hence Ʈ̃ is FSPN operator Theorem 3.14: Let Ʈ̃ ∈ �̃�(�̃� ) is a FSN then Ʈ̃∗̃ is a FSPN Proof: Since Ʈ̃ is fuzzy soft normal operator We know that Ʈ̃∗̃Ʈ ̃ =̃ Ʈ̃ Ʈ̃∗̃ if and only if ‖Ʈ̃∗̃ 𝑙𝜂ɠ(𝑒) ‖ ̃ =̃ ‖Ʈ̃ 𝑙𝜂ɠ(𝑒) ‖ ̃ For every unit vector 𝑙𝜂ɠ(𝑒) ∈̃ �̃� Let ‖Ʈ̃∗̃ 𝑙𝜂ɠ(𝑒) ‖ ̃ =̃ 〈Ʈ̃∗̃ 𝑙𝜂ɠ(𝑒) , Ʈ̃∗̃ 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃∗̃ 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃ Ʈ̃∗̃ 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 141 https://internationalpubls.com =̃ 〈(Ʈ̃∗̃Ʈ ̃)𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) ̃ 〉 =̃ 〈Ʈ̃2 𝑙𝜂ɠ(𝑒) , 𝑙𝜂ɠ(𝑒) 〉 ≤̃ ‖(Ʈ̃∗̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ ‖Ʈ̃∗̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖(Ʈ̃∗̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ ‖𝑙𝜂ɠ(𝑒) ̃‖ implies that ‖Ʈ̃∗̃𝑙𝜂ɠ(𝑒) ̃ ‖ 2 ≤̃ ‖(Ʈ̃∗̃) 2 𝑙𝜂ɠ(𝑒) ‖ ̃ Therefore, Ʈ̃∗̃ is FSPN IV Conclusion The ideas of normed space, metric space, and Hilbert space provide the soft and fuzzy updates. 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