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A Study of Paranormal Operators on Hilbert Spaces

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A Study of Paranormal Operators on Hilbert Spaces International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.5, No.8, August 2016 DOI:10.15693/ijaist/2016.v5i8.69-74 A study of paranormal operators on Hilbert Spaces Mr. P.J.Dowari Dr.N.Goswami Research Scholar/Department of Mathematics Asstt Professor/Department of Mathematics Tripura University, Agartala, India GauhatiUniversity, Guwahati, India Abstract- In this paper we have studied bounded bilinear functionals from 푋∗ × 푌∗ to 핂,and paranormal and n-paranormal operators on is defined by Hilbert spaces and their tensor product. (푥 ⊗ 푦)(푓, 푔) = 푓(푥)푔(푦) (푓 ∈ 푋∗, 푔 ∈ 푌∗) Considering two paranormal operators on a The algebraic tensor product of X and Y, 푋 ⊗ 푌is Hilbert space푯, we derive the condition under defined to be linear span of which their different operations are again {푥 ⊗ 푦 ∶ 푥 ∈ 푋, 푦 ∈ 푌 } 푖푛 퐵퐿(푋∗, 푌∗; 핂). (refer paranormal. Also, taking two Hilbert spaces 푿 to [4]) and 풀 and a paranormal operator 푻ퟏ on 푿 and 푻ퟐ on 풀 we define a paranormal operator 푻 on their Definition 2.Given normed spaces X and Y, the projective tensor product. Different properties of projective tensor norm 훾on 푋 ⊗ 푌is defined by this operator 푻 regarding paranormality are also discussed here. 훾 푢 = inf ∥ 푥푖 ∥∥ 푦푖 ∥: 푢 = 푥푖 ⊗ 푦푖 푖 푖 where the infimum is taken over all finite AMS subject classification: 47A05, 47A07, 47A10, representations of u. 47A12, 47A15 The completion of 푋 ⊗ 푌with respect to 훾is called Keywords: algebraic tensor product, projective the projective tensor product of 푋and 푌and is tensor norm, paranormal, 푀-paranormal, 푛- denoted by 푋 ⊗ 푌. paranormal operators. 훾 If 푋 and 푌are Hilbert spaces, an inner product on I. INTRODUCTION 푋 ⊗훾 푌is defined as The class of paranormal operators was introduced by 푎 ⊗ 푏, 푐 ⊗ 푑 = 푎, 푐 푏, 푑 V. Istratescu in 1960s, though the term “paranormal” where 푎, 푐 ∈ 푋 and 푏, 푑 ∈ 푌. is probably due to Furuta(1967). A paranormal Then it can be shown that 푋 ⊗훾 푌is a Hilbert space. operator is a generalization of a normal operator. Every hyponormal operator (in particular a quasi- Definition 3. An operator T on a Hilbert Space 퐻is normal operator and a normal operator) is said to be hyponormal if 푇∗ 푇 ≥ 푇 푇∗which is paranormal. Here we examine some properties of equivalent to ∥ 푇∗ 푥 ∥ ≤ ∥ 푇 푥 ∥, ∀푥 ∈ 퐻. paranormal operators and 푛-paranormal operators. An operator T is quasi hyponormal if (푇∗)2푇2 ≥ (푇∗푇)2 holds, which is equivalent to Definition 1.Let 푋, 푌be normed spaces over 핂 with ∥ 푇∗푇(푥) ∥≤∥ 푇2(푥) ∥, ∀푥 ∈ 퐻. ∗ ∗ dual spaces 푋 , 푌 . Given 푥 ∈ 푋, 푦 ∈ 푌, let 푥 ⊗ 푦 An operator T is quasi 푀-hyponormal if be the element of 퐵퐿(푋∗, 푌∗; 핂), the set of all ∥ 푇∗푇 푥 ∥≤ 푀 ∥ 푇2 푥 ∥, ∀푥 ∈ 퐻, 푀 > 0. Definition 4.An operator 푇 on a Hilbert Space 퐻is said to be paranormal if 2 2 ∥ 푇 푥 ∥ ≤∥ 푇 푥 ∥,for every unit vector 푥in 퐻. Another equivalent definition of paranormal operator in terms of topological aspects can be given as: 69 International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.5, No.8, August 2016 DOI:10.15693/ijaist/2016.v5i8.69-74 An operator 푇on a Hilbert Space 퐻is said to be Remark 1: If 푇is hyponormal then 푅푠푝 푇 = ∥ 푇 ∥ paranormal if ∥ 푇 − 푧퐼 −1 ∥ = 1/푑(푧, (푇)) for whereas, if 푇is paranormal then 푅푠푝(푇) ≤ ∥ 푇 ∥. all 푧 ∉ (푇) where 푑(푧, (푇)) is the distance from Remark 2.If 푇is hyponormal then 푇−1is also 푧 푡표 (푇),the spectrum of 푇. hyponormal, but these properties do not generalize to An operator 푇 on a Hilbert Space 퐻is said to be paranormal operators. 푀 −paranormal if, 푇 푥 ∥2 ≤ 푀 ∥ 푇2 푥 ∥,for II.MAIN RESULTS every unit vector 푥in 퐻and for some positive real First we consider the sum of two paranormal number 푀. operators on a Hilbert space 퐻. In general, the sum An operator 푇on a Hilbert Space 퐻 is said to be of two paranormal operators is not a paranormal 푛-paranormal (for some positive integer 푛) if operator. 푛 푛 푛−1 ∥ 푇 푥 ∥ ≤ ∥ 푇 푥 ∥∥ 푥 ∥ , ∀푥 ∈ 퐻. 1 1 −1 0 For example let 푇= and 푆 = An operator 푇is called polynomially paranormal if 0 1 0 −1 there exists a non-constant polynomial 푞(푧) such that be two operators on 2-dimensional space. 푞(푇) is paranormal. Then푇 and푆 are bothparanormal while푇 + 푆is not In general the following implication holds: so. normal ⇒ 푛-normal ⇒hyponormal ⇒paranormal⇒ We derive the conditions for which the sum will be a 푛-paranormal paranormal operator. But the converse is not true. Definition 5.For 푇 ∈ 훽(퐻), 푅(푇) and 푁(푇) denotes Theorem 1.If 푆 and 푇 are paranormal operators on a the range and the null space of 푇, respectively. Hilbert Space 퐻 then 푆 + 푇 is paranormal if An operator 푇 ∈ 훽(퐻) is said to have finite ascent if 푆∗퐴푆 ≥ 0푆∗퐵푇 ≥ 0푆∗퐴푇 ≥ 0 푁(푇푚 ) = 푁(푇푚+1) for some positive integer 푚, 푇∗퐵푇 ≥ 0푇∗퐴푆 ≥ 0푇∗퐵푆 ≥ 0 and finite descent if 푅(푇푚 ) ) = 푅(푇푚+1) for some where 퐴 = 푇∗푇 − 푇푇∗ 퐵 = 푆∗푆 − 푆푆∗. positive integer m. Proof.For 푆 + 푇to be paranormal, using lemma 8 we Lemma 6[9].If 푇 is paranormal, then 푇푛 is have, paranormal for any 푛 ∈ ℕ . (푆 + 푇)∗2 푆 + 푇 2 ≥ ((푆 + 푇)∗(푆 + 푇))2 Lemma 7 [6].A compact paranormal operator is ⇒ 푆∗ + 푇∗ 푆∗ + 푇∗ 푆 + 푇 푆 + 푇 2 normal. ≥ 푆 + 푇 ∗ 푆 + 푇 Lemma 8[2].푇is paranormalif 푇∗2푇2 ≥ 푇∗푇 2. ⇒ 푆∗2 + 푆∗푇∗ + 푇∗푆∗ + 푇∗2 푆2 + 푆푇 + 푇푆 + 푇2 ∗ ∗ ∗ ∗ Lemma 9[9]. If퐴 is any operator on 퐻, then there ≥ 푆 푆 + 푆 푇 + 푇 푆 + 푇 푇 (푆∗푆 + 푆∗푇 + 푇 푆∗ + 푇∗푇) exists a Hibert space 퐾 and a normal operator 푁 on 퐾 ⇒ 푆∗2푆2 + 푆∗2푆푇 + 푆∗2푇 푆 + 푆∗2푇2 + 푆∗푇∗푆2 such that 푇 = 퐴 ⊕ 푁 ∈ 훽(퐻 ⊕ 퐾) is + 푆∗푇∗푆푇 + 푆∗푇∗푇 푆 + 푆∗푇∗푇2 + 푇∗푆∗푆2 paranormal. +푇∗푆∗푆푇 + 푇∗푆∗푇 푆 + 푇∗푆∗푇2 + 푇∗2푆2 + 푇∗2푆푇 Lemma 10[9]There exists an invertible paranormal +푇∗2푇 푆 + 푇∗2푇2 operator 푇such that ≥ 푆∗푆푆∗푆 + 푆∗푆푆∗푇 + 푆∗푆푇∗푆 + 푆∗푆푇∗푇 1. 푇is not hyponormal. +푆∗푇 푆∗푆 + 푆∗푇 푆∗푇 + 푆∗푇 푇∗푆 + 푆∗푇 푇∗푇 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2. 푇2 is not paranormal. +푇 푆 푆 + 푇 푆푆 푇 + 푇 푆푇 푆 + 푇 푆푇 푇 + 푇 푇푆 푆 +푇∗푇푆∗푇 + 푇∗푇푇∗푆 + 푇∗푇푇∗푇 3. ∥ 푇 ∥ ≥ 푅 (푇); 푅 (푇) denotes spectral 푠푝 푠푝 ⇒ 푆∗2푆푇 + 푆∗푇∗푇푆 + 푆∗푇∗푇2 + 푇∗푆∗푆2 radius of 푇. +푇∗푆∗푆푇 + 푇∗2푇푆 4. 푇−1 is not paranormal. ≥ 푆∗푆푆∗푇 + 푆∗푇푇∗푆 + 푆∗푇푇∗푇 + 푇∗푆푆∗푆푇∗푆푆∗푇 +푇∗푇푇∗푆 + 푇∗푆푆∗푇 + 푇∗푇푇∗푆 Lemma 11.[9] If 푇푛is a sequence of paranormal ⇒ 푆∗2푆푇 + 푇∗2푇푆 + 푆∗푇∗푇 푆 + 푇 operators approaching the operator 푇 in norm, then + 푇∗푆∗푆 푆 + 푇 (푇푛 ) → (푇)as 푛 → ∞. ≥ 푆∗푇푇∗ 푆 + 푇 + 푇∗푆푆∗ 푆 + 푇 + 푆∗푆푆∗푇 Lemma 12.[2]Every quasi 푀-hyponormal + 푇∗푇푇∗푆 operator is paranormal. ⇒ 푆∗2푆푇 + 푇∗2푇푆 + 푆∗푇∗푇 + 푇∗푆∗푆 푆 + 푇 Lemma 13.[19]If푇 ∈ 훽(퐻) is polynomially ≥ 푆∗푆푆∗푇 + 푇∗푇푇∗푆 + 푆∗푇푇∗ + 푇∗푆푆∗ 푆 + 푇 paranormal, then푇 − 휆has finite ascent for all ∗2 ∗ ∗ 휆 ∈ ℂ. Now, 푆 푆푇 ≥ 푆 푆푆 푇 ⇒ 푆∗ 푆∗푆 − 푆푆∗ 푇 ≥ 0 70 International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.5, No.8, August 2016 DOI:10.15693/ijaist/2016.v5i8.69-74 푇∗2푇푆 ≥ 푇∗푇푇∗푆 푚푎푥{∥ 퐴 ∥2∥ 푥 ∥2, 2 ∥ 퐴 ∥∥ 퐵 ∥∥ 푥 ∥2, ∥ 퐵 ∥2 ∗ ∗ ∗ ⇒ 푇 푇 푇 − 푇푇 푆 ≥ 0 2 1 2 ∗ ∗ ∗ ∗ ∥ 푥 ∥ } ≤ ∥ A + B 푥 ∥ 푆 푇 푇 + 푇 푆 푆 푆 + 푇 2 ∗ ∗ ∗ ∗ ≥ 푆 푇푇 + 푇 푆푆 푆 + 푇 and 푇 = 퐴 + 퐵, then 푇 is paranormal. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 푆 푇 푇 + 푇 푆 푆 − 푆 푇푇 − 푇 푆푆 푆 + 푇 ≥ 0 Proof. The proof follows immediately. ∗ ∗ ∗ ∗ ∗ ∗ 푆 푇 푇 − 푇푇 + 푇 푆 푆 − 푆푆 푆 + 푇 ≥ 0 푆∗ 푇∗푇 − 푇푇∗ 푆 ≥ 0 Theorem 4.If 푇 is paranormal and 푆 is unitarily ⇒ 푆∗ 푇∗푇 − 푇푇∗ 푇 ≥ 0 푇∗ 푆∗푆 − 푆푆∗ 푆 ≥ 0 equivalent to 푇, then 푆 is paranormal. ⇒ 푇∗ 푆∗푆 − 푆푆∗ 푆 ≥ 0 Proof.For 푆 to be unitarily equivalent to 푇 Let, 퐴 = 푇∗푇 − 푇푇∗ 퐵 = 푆∗푆 − 푆푆∗ We have, 푆 = 푈푇푈∗ for some unitary operator 푈 The conditions are ⇒ 푆2 = 푈푇2푈∗ 푆∗퐴푆 ≥ 0푇∗퐵푇 ≥ 0 Now, ∥ 푆 푥 ∥2 = ∥ 푈푇푈∗ 푥 ∥2 푆∗퐴푇 ≥ 0 푆∗퐵푇 ≥ 0 = ∥ 푈 푇푈∗푥 ∥2 푇∗퐴푆 ≥ 0 푇∗퐵푆 ≥ 0 = ∥ 푇푈∗푥 ∥2(Since U is isometric) Then 푆 + 푇is paranormal. ≤ ∥ 푇2(푈∗푥) ∥. ∥ 푥 ∥ = ∥ 푈(푇2푈∗푥) ∥. ∥ 푥 ∥ For the product of two paranormal operators we get = ∥ 푈푇2푈∗푥 ∥. ∥ 푥 ∥ the following theorem. = ∥ 푆2푥 ∥. ∥ 푥 ∥ ⇒ ∥ 푆 푥 ∥2 ≤ ∥ 푆2푥 ∥. ∥ 푥 ∥ Theorem 2.If 푆 and 푇 are two commuting ∴ 푆is paranormal. paranormal operators on a Hilbert space 퐻, then 푇푆 is Theorem 5.If 푇 is a sequence of paranormal paranormal under the condition: 푛 operator and 푇 → 푇 then Tis paranormal.
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