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Generalized Paranormal Operators N Sivakumar N, Abirani Angelin S; International Journal of Advance Research, Ideas and Innovations in Technology. ISSN: 2454-132X Impact factor: 4.295 (Volume3, Issue2) Available online at www.ijariit.com Generalized Paranormal Operators N. Sivakumar S. Angelin Abirani Hindusthan College of Arts and Science, Tamil Nadu Hindusthan College of Arts and Science, Tamil Nadu [email protected] [email protected] Abstract: AN operator T ∈B (H) is said to be generalized P-paranormal if ퟏ ‖|푻|풑푼 |푻|풑풙 ‖‖풙‖ ≥ ‖|푻|풑풙‖ퟐ 푴풑 For all 풙 ∈ 푯, p > 0, and M > 0, where U is the partial isometry appeared in the polar decomposition 푻 = 푼|푻| of T. The aim of this note is to obtain some structure theorem for a class of generalized P-paranormal operators. Exactly we will give some conditions which are generalization of concepts of generalized paranormal operators. Keywords: Paranormal Operators, Hilbert Space, Hypo Normal Operator, Log-Hypo Normal Operators, and Bounded Linear Operator. INTRODUCTION Let H be an infinite dimensional complex Hilbert and B (H) denote the algebra of all bounded linear operators acting on H. Every operator T can be decomposed into T = U|푇|with a partial isometry U, where |푇| is the square root of T * T. If U is determined uniquely by the kernel condition ker (U) = ker|푇|, then this decomposition is called the polar decomposition, which is one of the most important results in operator theory. In this paper, T = U|푇| denotes the polar decomposition satisfying the kernel condition ker (U) = ker(|푇|). Recall that an operator T ∈B (H) is positive, T ≥ 0, if (T푥, 푥) ≥ 0 for all 푥 ∈ 퐻. An operator T ∈ B(H) is said to be hyponormal if T*T ≥ TT*. Hyponormal operators have been studied by many authors and it is known that hyponormal operators have many interesting properties similar to those of normal operators. An operator T is said to be p-hyponormal if (T ∗ 푇)푝 ≥ (푇푇 ∗)푝 for p ∈(0, 1] and an operator T is said to be log-hyponormal if T is invertible and log|푇| ≥ log |T*|. P-hyponormal and log- hyponormal operators are defined as extension of hyponormal operator. An operator T ∈B (H) is said to be paranormal if it satisfies the following norm inequality ‖푇2푥‖ ≥ ‖|푇푥‖2 For every unit vector 푥 ∈ 퐻. Ando [3] proved that every log-hyponormal operators is paranormal. It was originally introduced as an intermediate class between hyponormal operators and normaloid. It has been studied by many authors, so there are many to cite their references, for instance [3, 9, 22]. We say that an operator T belong to class A if |푇2| ≥ |푇|2. Class A was first introduced by Furuta-Ito-Yamazaki as a subclass of paranormal which include the class of p-hyponormal and log-hyponormal operators. DEFINITION 1.1 A bounded linear operator T on H is called generalized n-paranormal operator if for every unit vector 푥 ∈ 퐻, M > 0 and 푛 1 푛 a positive integer n such that 푛 ≥ 2, T satisfies ‖푇 푥‖ ≥ 푛 ‖푇푥‖ 푀2 DEFINITION 1.2 Let T ∈B(H), An operator T belongs to generalized class A operator if for M> 0, 푇 Satisfies 1 |푇2| ≥ |푇|2 푀 Rai [18] has defined a bounded operator T on a Hilbert space H as generalized paranormal if for every unit vector 푥 ∈ 퐻 and M> 0, 푇 Satisfies ‖푇2푥‖ ≥ 1 ‖푇푥‖2. He also proved a result for every unit vector 푥, 푀 1 ‖푇푘+1푥‖2 ≥ ‖푇푘푥‖2‖푇2푥‖ 푀2푘−1 © 2017, www.IJARIIT.com All Rights Reserved Page | 1065 Sivakumar N, Abirani Angelin S; International Journal of Advance Research, Ideas and Innovations in Technology. Where T is a bounded linear operator H, M>0 and k≥1 On the basis of the above result, we define the generalized n-paranormal operator as follows. THEOREM 1.3 2 1 If T satisfies |푇푛|푛 ≥ |T|2 for some positive integer n such that n≥2 and m>0, then T is a generalized n-paranormal operator. 푀 LEMMA 1.4 Holder – Mccarthy inequality Let T be a positive operator. Then the following inequalities hold for all 푥 ∈ 퐻. (i) < 푇푟푥, 푥 > ≤ < 푇푥, 푥 >푟 ‖푥‖2(1−푟) 푓표푟 0 < 푟 ≤ 1 (ii) < 푇푟푥, 푥 > ≥ < 푇푥, 푥 >푟 ‖푥‖2(1−푟) 푓표푟 푟 ≥ 1 Proof: 2 1 Suppose T satisfies |푇푛|푛 ≥ |T|2 -------- (B-1) 푀 for some positive integer n such that n≥2 and m>0, Then for every unit vector 푥 ∈ 퐻. ‖푇푛‖2 = < |푇푛|2푥, 푥 > 2 푛 푛 ≥ < |푇 |푛푥, 푥 > 1 2 푛 ≥ < |푇| 푥, 푥 > ---- (B-1) 푀 1 ≥ < |푇|2푥, 푥 >푛 푀푛 ≥ 1 ‖푇푥‖2푛 푀푛 푛 1 푛 Hence we have‖푇 ‖ ≥ 2 ‖푇푥‖ for every unit vector 푥 ∈ 퐻 푀푛 THEOREM 1.5 Let T ∈B(H). If T is generalized k -quasi – hyponormal then T is generalized (k+1) -paranormal. Proof: If T is a generalized k – quasi hyponormal then the following relation holds for every unit vector 푥 ∈ 퐻 푘+1 1 ∗ 푘 ‖푇 푥‖ ≥ 푘+1 ‖푇 푇 푥‖ ---- (B-1) ( ) 푀 2 To prove T is generalized (K+1) -paranormal it suffices to prove that 푘+1 1 푘+1 ‖푇 푥‖ ≥ 푘+1 ‖푇푥‖ ( ) 푀 2 We know that for any bounded linear operator T on a Hilbert space H. ‖푇푥‖푘+1 ≤ ‖푇∗푇푘푥‖ ---------- (B-3) Therefore from (B-2) and (B-3) we get 푘+1 1 푘+1 ‖푇 푥‖ ≥ 푘+1 ‖푇푥‖ ( ) 푀 2 Hence T is a generalized (k+1) -paranormal operator. THEOREM 1.6 Let 0<p<1. Every Generalized p-paranormal operator is generalized paranormal. Proof: We note that the Holder inequality by Mccarthy (i) of Lemma 1.4 has the following form. ‖푆푝푦‖ ≤ ‖푆푦‖푝‖푦‖1−푝 For all 푦 ∈ 퐻. Putting 푆 = |푇| and 푦 = 푈|푇|푝in part (ii) of Lemma 1.4 We have ‖|푇|푝푈 |푇|푝푥‖ ≤ ‖|푇| 푈|푇|푝푥‖푝‖|푇|푝푥 ‖1−푝 Since the left hand side of the above inequality is greater than ‖|푇|푝푥‖2 / 푀푝‖푥‖by the generalized absolute p-paranormality. It follows that ‖|푇|푝푥‖1+푝 ≤ ‖|푇| 푈|푇|푝푥‖푝‖푥‖ --------- (B.4) Hence, if we replace 푥 by |푇|1−푝푥 in (B.4) then 1 푝+1 1−푝 2 푝 ‖푇푥‖ ≤ ‖|푇| 푥‖‖푇 푥‖ 푀푝 Applying part (ii) of lemma 1.4 again it follows that ‖|푇|1−푝푥‖ ≤ ‖푇푥‖1−푝‖푥‖푝 Therefore it implies that 1 푝+1 1−푝 2 푝 ‖푇푥‖ ≤ ‖|푇| 푥‖‖푇 푥‖ 푀푝 ≤ ‖푇푥‖1−푝‖푥‖푝‖푇2푥‖푝 so that 1 2 2 ‖푇푥‖ ≤ ‖푇 푥‖‖푥‖ 푀 This completes the proof. © 2017, www.IJARIIT.com All Rights Reserved Page | 1066 Sivakumar N, Abirani Angelin S; International Journal of Advance Research, Ideas and Innovations in Technology. THEOREM 1.7 Let T be a generalized p-paranormal operator, then ‖푇3푥‖ ≥ 1 ‖푇2푥‖‖푇푥‖For every unit vector 푥 ∈ 퐻 푀2 Proof: For a unit vector 푥 in H. We may assume that ‖푇푥‖ ≠0, we have 3 2 푇푥 ‖푇 푥‖=‖푇푥‖ ‖푇 ‖ ‖푇푥‖ 1 푇푥 2 ≥ ‖푇푥‖ ‖푇 ‖ (By theorem 1.6) 푀 ‖푇푥‖ 2 1 푇2푥 ≥ ‖푇푥‖ ‖ ‖ 푀 ‖푇푥‖ 2 1 ‖푇푥‖‖푇2푥‖ ≥ 푀 ‖푇푥‖2 1 ‖푇푥‖‖푇2푥‖ ‖푇푥‖2 ≥ 푀2 ‖푇푥‖2 1 ‖푇3푥‖ ≥ ‖푇2푥‖‖푇푥‖ 푀2 Hence the theorem. THEOREM 1.8 Let T be a generalized p-paranormal operator, then 4 1 2 2 ‖푇 푥‖ ≥ ‖푇 푥‖ ‖푇푥‖For every unit vector 푥 ∈ 퐻 푀5 Proof: For a unit vector 푥 in H. We may assume that ‖푇푥‖ ≠0, we have 2 ‖푇4푥‖=‖푇푥‖ ‖푇2 푇 푥 ‖ ‖푇푥‖ 2 1 푇2푥 ≥ ‖푇푥‖ ‖푇 ‖ (By theorem 1.6) 푀 ‖푇푥‖ 2 1 ‖푇푥‖‖푇3푥‖ ≥ 푀 ‖푇푥‖2 1 ‖푇푥‖‖푇3푥‖ ‖푇3푥‖ ≥ 푀 ‖푇푥‖2 2 ‖푇2푥‖ ‖푇푥‖2 ≥ 1 (By Theorem 1.7) 푀5 ‖푇푥‖ 1 ‖푇4푥‖ ≥ ‖푇2푥‖2‖푇푥‖ 푀5 THEOREM 1.9 Let T be a generalized P-Paranormal operator, then ‖푇푘+1푥‖2 ≥ 1 ‖푇푘푥‖2‖푇2푥‖ For a positive integer k≥ 1 and every unit vector 푥 in H. 푀2푘−1 Proof: We will use induction to establish the inequality. ‖푇푘+1푥‖2 ≥ 1 ‖푇푘푥‖2‖푇2푥‖ For a positive integer k≥ 1 ...... (B.5) 푀2푘−1 In case K=1 ‖푇2푥‖2 = ‖푇2푥‖ ‖푇2푥‖ ≥ 1 ‖푇푥‖4........... (B.6) 푀2 hold by theorem (2.4). Now suppose that (B.5) holds for some k≥ 1 and we assume that ‖푇푥‖ ≠ 0 then 푇푥 2 ‖푇푘+2푥‖2 = ‖푇푥‖2 ‖푇푘+1 ‖ ‖푇푥‖ 2 2 1 푘 푇푥 2 푇푥 ≥ ‖푇푥‖ ‖푇 ‖ ‖푇 ‖ (By B.5) 푀2푘−1 ‖푇푥‖ ‖푇푥‖ 푘+1 3 2 1 푇 푥 ‖푇 푥‖ ≥ ‖푇푥‖ ‖ ‖ 푀2푘−1 ‖푇푥‖ ‖푇푥‖ 2 1 푘+1 2 1 ‖푇 푥‖‖푇푥‖ ≥ ‖푇 푥‖ 푀2푘−1 푀2 ‖푇푥‖ 1 푘+1 2 2 ≥ ‖푇 푥‖ ‖푇 푥‖ 푀2푘+1 That is ‖푇푘+2푥‖2 ≥ 1 ‖푇푘+1푥‖2‖푇2푥‖ ......... (B.7) 푀2푘+1 For k+3 푇푥 2 ‖푇푘+3푥‖2 = ‖푇푥‖2 ‖푇푘+2 ‖ ‖푇푥‖ 2 2 1 푘+1 푇푥 2 푇푥 ≥ ‖푇푥‖ ‖푇 ‖ ‖푇 ‖by (B.7) 푀2푘+1 ‖푇푥‖ ‖푇푥‖ © 2017, www.IJARIIT.com All Rights Reserved Page | 1067 Sivakumar N, Abirani Angelin S; International Journal of Advance Research, Ideas and Innovations in Technology. 푘+2 2 3 2 1 푇 푥 푇 푥 ≥ ‖푇푥‖ ‖ ‖ ‖ ‖ 푀2푘+1 ‖푇푥‖ ‖푇푥‖ 푘+2 2 3 2 1 ‖푇 푥‖ ‖푇 푥‖ ≥ ‖푇푥‖ 푀2푘+1 ‖푇푥‖2 ‖푇푥‖ 2 1 푘+2 2 1 ‖푇 푥‖‖푇푥‖ ≥ ‖푇 푥‖ (By theorem 1.7) 푀2푘+1 푀2 ‖푇푥‖ ‖푇푘+3 푥‖2 ≥ 1 ‖푇푘+2 푥‖2‖푇2푥‖.......
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