Research article · 2022 · Full text
International Journal of Statistics and Applied MathematicsOn metric equivalence of operators of order N
Volume 7(1) · Pages 94–97 · DOI: 10.22271/maths.2022.v7.i1b.777 ↗
Research areasAdvanced Topics in AlgebraHolomorphic and Operator TheoryAdvanced Algebra and Logic
Abstract
Author-writtenClass of operators satisfying V*nVn= Q*nQn is introduced which is an equivalence relation, that is Metric equivalence of order n. We examine some properties that this class enjoys. On the same note, we study the relation of this class to other classes of operators like K* quasinormal operators of order n and normal of order n. We give a condition under which V*nVn= Q*nQn is same as V*Vn= Q*Qn and V*nVm= Q*nQm, that is n-power and (n,m)-power-Metric equivalences respectively.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- The study focuses on the behavior of various classes of mathematical operators, building on previous research into n-normal, K*-quasi-normal, and quasi-normal operators. View evidence ↗
- Prior work has established the properties of (n, m)-metrically equivalent and n-metrically equivalent operators and their relationships to other operator classes. View evidence ↗
The question
- The authors aim to introduce and examine the properties of a class of operators satisfying the relation V*nVn= Q*nQn, termed 'Metric equivalence of order n'. View evidence ↗
- The research seeks to determine how metric equivalence of order n relates to other operator classes, specifically normal operators of order n and K*-quasinormal operators of order n. View evidence ↗
What they did
- The authors define metric equivalence of order n for two operators V and Q in L(H) as the condition where V*nVn = Q*nQn for any positive integer n. View evidence ↗
- The study utilizes the definition of a normal operator of order n as one where V*nVn = VnV*n. View evidence ↗
- The authors analyze conditions under which metric equivalence of order n is equivalent to n-power and (n, m)-power-metric equivalences. View evidence ↗
What they found
- An operator Q is a normal operator of order n if and only if Q and its adjoint Q* are metrically equivalent of order n. View evidence ↗
- Isometric operators that are unitarily metrically equivalent of order 2+n are also metrically equivalent of order n. View evidence ↗
- Quasi-isometric operators that are metrically equivalent of order 2 are also metrically equivalent. View evidence ↗
- Quasi-isometric operators that are metrically equivalent of order 4 are also metrically equivalent of order 3. View evidence ↗
What it means
- The authors conclude that the relation V*nVn= Q*nQn constitutes a valid equivalence relation, termed metric equivalence of order n, and have successfully characterized several of its properties. View evidence ↗
Visual summary
1 visualsIllustrations explain the paper; they do not show study data.
Operator Equivalence Concept
AI-generated conceptual illustration (Abstract geometric representations of mathematical operators). It explains the idea and does not show study data.
View source evidence ↗Introduction
We take into consideration behaviors of various classes of operators. Properties of n-normal operators was intensively explored [2] , K * quasi-normal operators of order n was covered in [5] and quasi-normal operators [1] . Wanjala Victor and A.M. Nyongesa [8] explored properties of (n, m)-Metric equivalence of operators in relationship between (n, m)-metrically equivalent operators and other classes like (n, m)-class (Q) and n-metrically equivalent operators were covered by Wanjala Victor and A.M. Nyongesa [8] . N-metrically equivalent as a new equivalence relation was introduced and sheltered by Wanjala Victor et al. 2020 [7] . Wanjala Victor et al. 2020 [7] striked results relating the class of n-metrically equivalent operators to supplementary classes such as quasinormal and k-quasinormal operators. [6] An operator V ∈L (H) is said to be normal operator of order n if V *n V n = V n V *n . Definition 2.1.2. [7] Operators V ∈L(H) and Q ∈L(H) are presumablyn-metrically equivalent, denoted by V~n -m Q, provided V * V n = Q * Q n for any positive integer n ∈ ℕ. Definition 2.1.3. [8] Operators V ∈L (H) and Q ∈L (H) are presumably (n, m)-metrically equivalent, provided V *m V n = Q *m Q n for any positive integer n, m ∈ ℕ. Definition 2.1.4. [4] Two operators V ∈L(H) and Q ∈L(H) are presumablymetrically equivalent, denoted by V~mQ, provided V * V= Q * Q. Definition 2.1.5. Two operators V ∈L(H) and Q ∈L(H) are presumably metrically equivalent of order n, denoted by V~m n Q, provided V *n V n = Q *n Q n for any positive integer n.
Preliminaries Definition 2.1.1:
Preliminaries Definition 2.1.1:
Main results
Theorem 3.1.1. Let V ∈L (H)bea normal operator of order n and Q∈L (H)beunitarily equivalent to V, it follows Qis a normal of order n. Proof: Since Q = U * VU and U being a unitary and V normal operator of order n, we have:
Hence the proof. Corollary 3.1.2. Operator Q∈L (H) is a normal of order n if and only if Q and Q * are metrically equivalent of order n. Proof. The proof trickles down from Theorem 2.1.1 above. Proposition 3.1.3. Suppose V ∈L (H) and Q∈L (H) are metrically equivalent of order n, it follows V * and Q * are co-metrically equivalent of order n. Proof. Since V and Q are metric equivalence relation of order n, we have;
Thus V * and Q * are co-metric equivalence of order n. Proposition 3.1.4. Let V ∈L (H) and Q ∈L (H) be metric equivalence relation of order n+q, it follows V and S are metric equivalence relation of order n+q+1and thus V and Q are metric equivalence of order n+k for every k≥q for all n, k, q which are reals. Proof. V and Q are metric equivalence of order n+q, we have;
Pre-multiplying by V * and Q * on the left-hand side and post-multiplying (F , ) by V and by Q on the right-hand side;
Thus V and Q are metrically equivalent operators of order n+q+1.
Relationship between metric equivalence of order n+q and other classes of operators Definition 4.1.1:
V ∈L (H) is presumably normal operator of order n+q for n, q > 0 if V *n+q V n+q = V n+q V *n+q Definition 4.1.2: S, T ∈L (H) are presumably metrically equivalent of order n+q if V *n+q V n+q = Q n+q Q *n+q for n, q > 0.
Remark 4.1.3: We note that normal operator of order n+q is V *n V n = V n V *n whenever q =1 and it's a normal operator whenever n=q=1. Proposition 4.1.4: LetV ∈L (H)and Q ∈L (H)beboth unitarilymetrically equivalent operators of order n+q+1 and metrically equivalent, tit follows V and Q are metrically equivalent of order n+q for n, q > 0, and if Vis normal operator of order n+q, it follows Qisnormal operator of order n+q for all n, q > 0. Proof. V and Q being unitarily metrically equivalent of order n+q+1 implies; V *(n+q+1) V n+q+1 = UQ *(n+q+1) Q n+p+1 U * ; and from Proposition 2.2.4, V and Q are metrically equivalence of order n+k since they are metrically equivalent of order n+q+1.Hence V *(n+q+1) V n+q+1 = UQ *(n+q+1) Q n+q+1 U * gives us;
From ( 3) and (4) we have; = UQ *(n+q) Q n+q U * = U Q n+q Q *(n+q) U * = Q *(n+q) Q n+q =Q n+q Q *(n+q) Thus Q is a normal operator of order n+q.
Hence, V and Q are metrically equivalent of order n+1 and the result follows by induction.
References
7 in the original paper- Alzuraiqi SA. On n-normal operators, General math. Notes. 2010;1(2):61-73.
- Senthil Kumar D, Masheswari Naik P, Santhi R. K-quasi-normal operators. International Journal of Mathematics and Computation. 2012;15(2):0974-5718.
- Nzimbi BM, Pokhariyal GP, Moindi SK. A note on Metric Equivalence of some operators, Far East Journal of Mathematical Sciences. 2013;75(2):301-318.
- Abdul Ali N, Prasanna A, Mathura G, Keerthiga A, Gayathri T. On k* Quasi Normal Operator of Order n, International Journal of Mathematics Trends and Technology. 2019;65(8):2231-5373.
- Shaakir Laith K, Marai Saad. Quasi-normal operator of order n. Tikrit Journal of Pure Sciences. 2010. ISSN 1662-1813.
- Wanjala Victor, Obogi RK, Okoya M. N-metric equivalence of operators, International Journal of mathematics and its applications. 2020;8(1):107-109.
- Wanjala Victor, Nyongesa AM. On (n, m)-metrically equivalent operators, International Journal of mathematics and its applications. 2021;9(2):101-104.