International Research Journal of Engineering and Technology (IRJET)
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Volume: 11 Issue: 02 | Feb 2024
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The class 𝑫(𝑻𝒌 ) −operators Shaymaa Shawkat Al-shakarchi Department of Mathematics, College of Basic Education, University of Kufa, Najaf, Iraq --------------------------------------------------------------------------***------------------------------------------------------------------Abstract: This article presents a new category of operators called 𝐷(𝑇 𝑘 ) − operators, which operate on a complex Hilbert space H. An operator 𝑇 ∈ 𝐵(𝐻) is considered a 𝐷(𝑇 𝑘 ) − operators if the equation (𝑇 ∗ 𝑇 )𝑘 𝑈 = 𝑈(𝑇 ∗ 𝑇)𝑘 holds, where 𝑘 is a positive integer greater than 1, and 𝑇 ∗ is the adjoint of the operator 𝑇, We will explore the fundamental characteristics of these operators and provide examples for better understanding. Keywords: Hilbert space, normal operators, 𝑫(𝑻) −operators, adjoint operators, bounded linear operators.
1-Introduction In this article, 𝐵(𝐻) refers to the algebra consisting of all bounded linear operators on a complex Hilbert space 𝐻. A Hilbert space is a mathematical space that has an inner product and is also characterized by its completeness with respect to the norm induced by this inner product. An operator 𝑇 ∗ is defined as the adjoint of 𝑇 if and only if the inner product (𝑇𝑥, 𝑦) is equivalent to (𝑥, 𝑇 ∗ 𝑦) for all 𝑥 and 𝑦 belonging to the set 𝐻. A normal operator , which maps from a Hilbert space 𝐻 to itself, is defined by the equation 𝑇 ∗ 𝑇 = 𝑇𝑇 ∗ . The theory of operators in Hilbert space has been thoroughly analysed by several authors, as seen by the references [1, 2, 3]. In 2021, Elaf. S. A. conducted an examination of the class of operators 𝐷(𝑇) as described in [4], and presented the fundamental characteristics of this class in a Hilbert space. An operator 𝑇 ∈ 𝐵(𝐻) is referred to as a 𝐷(𝑇) −operators if there exists 𝑈 ∈ 𝐵(𝐻) such that 𝑈 , and 𝑇 ∗ 𝑇𝑇 = 𝑇𝑇 ∗ 𝑇. This article presents the class 𝐷(𝑇 𝑘 ) − operators as an extension of the class 𝐷(𝑇) operators and examines its essential characteristics. We shall establish the conditions under which an operator T is considered to be 𝐷(𝑇 𝑘 ). To be classified in this category, the addition and multiplication of two operators 𝐷(𝑇 𝑘 ) must meet certain conditions, which we will examine. The representation of any operator 𝑇 in cartesian form is 𝑇 = 𝐴 + 𝐵𝑖 , where 𝐴 and 𝐵 represent the real and imaginary components of 𝑇, respectively. The real component of 𝑇, represented as 𝑅𝑒 𝑇, is calculated as the average of 𝑇 and its complex conjugate, which is
𝑇+𝑇 ∗ 2
.On the other hand, the imaginary component of 𝑇, written as 𝑚 𝑇, is determined by
taking the differences between 𝑇 and its complex conjugate, divided by 2𝑖, resulting in
𝑇−𝑇 ∗ 2𝑖
.
2-Main Results The objective of the work is to introduce a new category of operators known as 𝐷(𝑇 𝑘 ) − operators and analyze fundamental properties of this category. 2-1 Definition: Let 𝑇 be a bounded operator from a complex Hilbert space 𝐻 to itself, the 𝑇 is said to be a class 𝐷(𝑇 𝑘 ) − operators if there 𝑘
𝑘
exists is an operator 𝑈 ∈ 𝐵(𝐻) such that 𝑈 (𝑇 ∗ 𝑇 𝑘 ) = (𝑇 ∗ 𝑇 𝑘 ) 𝑈, where 𝑘 is a positive integer greater than 1.
2-2 Example: The operators 𝑇 and 𝑈 are two operators in the two-dimensional Hilbert space ₵2 , 2𝑖 2 −𝑖 where 𝑇 = * + and 𝑈 = * + −2𝑖 𝑖 1 2
𝑈(𝑇 ∗ 𝑇 2 ) = * 4𝑖
−4𝑖 +=* 4 4𝑖
2 −4𝑖 + = (𝑇 ∗ 𝑇 2 )𝑈. 4
So 𝑇 ∈ 𝐷(𝑇 2 ) − operators.
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