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:

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, 𝜎(푇)) for whereas, if 푇is paranormal then 푅푠푝(푇) ≤ ∥ 푇 ∥. all 푧 ∉ 𝜎(푇) where 푑(푧, 𝜎(푇)) is the distance from Remark 2.If 푇is hyponormal then 푇−1is also 푧 푡표 𝜎(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 𝜎(푇푛 ) → 𝜎(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

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. 푚푖푛 ∥ 푆2 ∥ 푇 ∥2 푥 ∥, ∥ 푇2 ∥ 푆 ∥2 푥 ∥ 푛 2 2 ≤ ∥ 푆2푇2푥 ∥ ∀푥 ∈ 퐻 . Proof.∥ 푇 푥 ∥ = 푙푖푚 ∥ 푇푛 푥 ∥ 2 Proof.For 푆to be paranormal we have, ≤ 푙푖푚 ∥ 푇푛 푥 ∥. ∥ 푥 ∥ 2 ∥ 푆 푥 ∥2 ≤ ∥ 푆2(푥) ∥. ∥ 푥 ∥ = ∥ 푇 푥 ∥. ∥ 푥 ∥ ∥ 푇 푥 ∥2 ≤ ∥ 푇2 (푥) ∥. ∥ 푥 ∥ i.e., 푇is paranormal.  ∥ 푆(푇푥) ∥2 ≤ ∥ 푆2(푇푥) ∥. ∥ 푇푥 ∥ Theorem 6. If 퐴 is any operator on a Hilbert space = ∥ 푇 푆2푥 ∥. ∥ 푇푥 ∥ 퐻, then there exists a Hilbert space 퐾and a normal 1 1 operator 푁on K such that 2 2 2 2 2 2 ≤∥ 푇 (푆 푥) ∥ . ∥ 푆 푥 ∥ . ∥ 푇 ∥. ∥ 푥 ∥∥ 푆 푇푥 ∥ =∥ 푇 = 퐴 ⊕ 푁 ∈ 훽(퐻 ⊕ 퐾)is푀 paranormal, 2 푇 푆푥 ∥ 푀 < 1. 2 ≤∥ 푇 푆푥 ∥. ∥ 푆푥 ∥ Proof.Since the numerical range, 푊(퐴) is a compact 1 1 ≤∥ 푆2(푇2푥) ∥2 . ∥ 푇2푥 ∥2 . ∥ 푆 ∥. ∥ 푥 ∥ set of complex numbers, there exists a Hilbert Thus, SpaceK and a Normal operator 푁on 퐾such that 1 1 ∥ 푆 푇푥 ∥2≤ min⁡[∥ 푇2 푆2푥 ∥2 . ∥ 푆2푥 ∥2 . ∥ 푇 ∥. 𝜎(푁) = 푊(퐴). 1 1 Let 푇 = 퐴 ⊕ 푁 2 2 2 ∥ 푥 ∥, ∥ 푆 (푇 푥 ∥2 . ∥ 푇 푥 ∥2 . ∥ 푆 ∥. ∥ 푥 ∥] 1 1 ⇒ 𝜎 푇 = 𝜎 푁 = 푊 퐴 =∥ 푆2푇2푥 ∥2 min ∥ 푇 ∥. ∥ 푆2푥 ∥2, ∥ 푆 ∥ Let 푧 ∉ 𝜎 푇 ⇒ 푧 ∉ 푊 퐴 1 1 ∥ 푇2푥 ∥2 . ∥ 푥 ∥ ∴ ∥ 푅 퐴, 푧 ∥ ≤ 푠푝 푑(푧, 푊(퐴)) 1 1 ≤∥ 푆2푇2푥 ∥2. ∥ 푆2푇2푥 ∥2. ∥ 푥 ∥ 1 = =∥ 푆2푇2푥 ∥. ∥ 푥 ∥ 푑(푧, 𝜎(푇)) 1 ∴ 푆푇 is paranormal.  ≤ (for푀 < 1) 푀푑 푧,𝜎 푇

Theorem 3.If 퐴 and 퐵 are any two bounded ∵ 푁is normal and 푧 ∈ 𝜌 푁 . operators on a Hilbert space 퐻 satisfying 1 ∥ 푅 푁, 푧 ∥ = 푠푝 푀푑 푧, 𝜎 푇

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

∥ 푅푠푝 푇, 푧 ∥ = max⁡{∥ 푅푠푝 퐴, 푧 ∥, ∥ 푅푠푝 푁, 푧 ∥} [∵ 푇is paranormal ] 1 2 2 = =∥ 푇1 푥 ⊗ 푇2 푚2 ∥ 푀푑(푧, 𝜎(푇)) =∥ 푇2(푥) ∥∥ 푇2(푚 ) ∥ ∴ 푇is푀-paranormal.  1 2 2 2 2 Theorem 7.푇 is n-paranormal if and only if ∥ ⇒∥ 푇1 푥 ∥ ≤∥ 푇1 푥 ∥ 푇 푥 ∥2 − 2휆 ∥ 푇푥 ∥푛 + 휆2 ≥ 0, with ∥ 푥 ∥ = 1 and 푛 showing that 푇1 is paranormal. 휆 ≥ 0. Proof.We know for positive numbers 푏and 푐, Similarly we can also show that 푇2 is paranormal.  푐 − 2푏휆 + 휆2 ≥ 0 ∀ 휆 ≥ 0if and only if Proceeding in the above way we can prove the 푏2 ≤ 푐. following results. Let 푏 = ∥ 푇 푥 ∥푛 ; 푐 = ∥ 푇푛 푥 ∥2; ∥ 푥 ∥ = 1 . T is paranormal if and only if ∥ 푇푥 ∥푛 ≤ ∥ 푇푛 푥 ∥. Theorem 9.a 푇 is 푀-paranormal (푀 < 1)if 푇1and Therefore, 푇is paranormal if and only if ∥ 푇푛 푥 ∥2 − 푇2are 푀-paranormal. Conversely if 푇 is 푀- 2휆 ∥ 푇푥 ∥푛 + 휆2 for each λ ≥ 0 with∥ 푥 ∥= 1. paranormal and 푇1 and 푇2 satisfy the condition as in Theorem 8(b), then 푇1 and 푇2 are 푀-paranormal. b. The above result holds for 푛-paranormality condition also. III.Some properties of paranormal operators on the tensor product of Hilbert Spaces. In the next theorem we replace 푇2 by a projection 푝 We construct an operator 푇on the projective tensor of the Hilbert space 푌 onto a closed subspace 퐾 of 푌 product of Hilbert spaces X and Y. Let T1 be an so 푇 ∶ 푋 ⊗훾 푌 → 푋 ⊗훾 푌 is such that operator on X and 푇2 be an operator on 푌. We define 푇 훴푖 푥푖 ⊗ 푦푖 = 훴푖 푇1 푥푖 ⊗ 푝 푦푖 푇 ∶ 푋 ⊗훾 푌 → 푋 ⊗훾 푌by

푇 훴푖 푥푖 ⊗ 푦푖 = 훴푖 푇1 푥푖 ⊗ 푇2 푦푖 ; Theorem 10.If 푇1 and 푇2 are quasi M- Σ푖푥푖 ⊗ 푦푖 ∈ 푋 ⊗훾 푌 hyponormaloperators on a closed subset 퐽 of 푋 then

Theorem 8.a.푇 is paranormal if 푇1 and 푇2 are 푇 is paranormal on 퐽 ⊗ 퐾. paranormal. ∗ Proof.Now, ∥ 푇 – 휆 푇 훴 푗푖 ⊗ 푘푖 ∥ b. If there exists points 푚 and 푚 such that ∥ ∗ 1 2 = ∥ 훴푖 푇1 − 휆 푇1푗푖 ⊗ 푝 푘푖 ∥ 2 2 ∗ 푇1 푚1 ∥ =∥ 푇1 푚1 ∥ and ≤ 훴푖 ∥ 푇1 − 휆 푇1푗푖 ∥∥ 푝(푘푖 ) ∥ 2 2 ∥ 푇2 푚2 ∥ =∥ 푇2 푚2 ∥and푇 is paranormal then푇1 ≤ 푀훴푖 ∥ 푇1 − 휆 푇1푗푖 ∥∥ 푝 푘푖 ∥ and 푇2 are paranormal. (Since 푇1 is quasi 푀-hyponormal) Proof.a Taking the projective tensor norm, 2 2 ∗ ∥ 푇 훴푖 푥푖 ⊗ 푦푖 ∥ =∥ 훴푖 푇1 푥푖 ⊗ 푇2 푦푖 ∥ ∥ 푇 − 휆 푇 훴 푗푖 ⊗ 푘푖 ∥ 2 2 ≤ 훴푖 ∥ 푇1(푥푖 )) ∥ . ∥ 푇2 푦푖 ∥ ≤ 푀 ∥ 훴푖 푇1 − 휆 푇1푗푖 ⊗ 푝(푘푖 ) ∥ 2 2 = 푀 ∥ 푇 − 휆 푇 훴 푗 ⊗ 푘 ∥ ≤ 훴푖 ∥ 푇1 푥푖 ∥. ∥ 푇2 푦푖 ∥ 푖 푖 Showing that 푇 is quasi 푀-hyponormal. [∵ 푇1and푇2are paranormal] 2 2 2 Hence by lemma 12, the proof follows.  Thus, ∥ 푇 푖 푥푖 ⊗ 푦푖 ∥ ≤ 훴푖 ∥ 푇1 푥푖 ⊗ 푇2 푦푖 ∥ [Taking projective norm] Theorem 11.If 푇1 is polynomially paranormal then 2 =∥ 푇 훴푖 푥푖 ⊗ 푦푖 ∥ 푇 − 휆 has finite ascent on 푋 ⊗ 퐾 for all 휆 ∈ ℂ. So, T is paranormal.  Proof.Since 푇1 is polynomially paranormal so by

lemma 13, 푇1 − 휆 has finite ascent on 푋 for all b. let푇be paranormal and푥 ∈ 푋be arbitrary. 휆 ∈ ℂ. 푝 푝+1 Now, ∴ 푁 (푇1 − 휆 ) = 푁( (푇1 − 휆 ), for some 2 2 2 positive integer 푝. ∥ 푇1 푥 ∥ ∥ 푇2 푚2 ∥ =∥ 푇1 푥 ⊗ 푇2 푚2 ∥ 2 Let푢 = 훴푖푥푖 ⊗ 푦푖 ∈ 푋 ⊗ 퐾be such that =∥ 푇 푥 ⊗ 푚2 ∥ 푢 ∈ 푁 푇 − 휆 푝 . ≤∥ 푇2 푥 ⊗ 푚 ∥ 2 ∴ (푇 − 휆)푝(푢) = 0

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

푝 ⇒ (푇 − 휆) (훴푖 푥푖 ⊗ 푘푖 ) = 0 [7] DoughlasR.G.”OnMajorization,Factorisation and 푝 ⇒ 훴푖 (푇1 − 휆 푥푖 ⊗ 푘푖 = 0 Range Inclusion of Operators on Hilbert [∵ 푘푖 ∈ 퐾, so, 푝 푘푖 = 푘푖 ] 푝 Spaces”,Pacific J. Math.28,1965, pp. 413-415. ⇒ 푥푖 ∈ 푁 푇1 − 휆 or푘푖 = 0∀푖 푝+1 [8] Dowari P.J. & N. Goswami ”Some Results on n- ⇒ 푥푖 ∈ 푁 푇1 − 휆 or 푘푖 = 0 ∀푖 푝+1 ⇒ 푇1 − 휆 푥푖 ⊗ 푘푖 = 0 ∀푖. Normal and Hyponormal Operators on the Tensor 푝+1 ⇒ 훴푖 푇1 − 휆 푥푖 ⊗ 푘푖 = 0 Product of Hilbert Spaces”, International J.of Math. 푝+1 ⇒ 푇 − 휆 훴푖 푥푖 ⊗ 푘푖 = 0 푝+1 Research;Vol. 7, No. 2; 2015,pp.141-149. ⇒ 훴푖 푥푖 ⊗ 푘푖 ∈ 푁( 푇 − 휆 ) Thus, 푁 푇 − 휆 푝 ⊆ 푁 푇 − 휆 푝+1 . [9] Glenn R.L.”Topological Properties of Similarly we can showthat Paranormal Operators On Hilbert 푁 푇 − 휆 푝+1 ⊆ 푁 푇 − 휆 푝 . Space”,Transcations of American Mathematical Therefore 푇 − 휆has finite ascent. Society,Vol.172,October,1972,pp 35-42 Remark: In [6], [19], Cho. M.et.al. and Z. Lingling [10] IstratescuV.”On some discussed about *푛-paranormal operators. HyponormalOperators”Pacific J. Math, Vol.,22, No. Let 푋be a complex Banach space. Let 훱(푋) = 3,1967. { 푥, 푓 ∈ 푋 × 푋∗: ∥ 푓 ∥ = 푓 푥 , ∥ 푥 ∥ = 1}, where 푋∗is the dual space of푋. [11] Jibril. A. A. S.,On 2-normal Operators Disarat, 푇 ∈ 훽(푋)is said to be *-푛-paranormalif Vol. 23, 1996. ∗ 푛 푛 ∥ 푇 푓 ∥ ≤ ∥ 푇 푥 ∥ for all (푥, 푓) ∈ 훱(푋), where [12] Jibril A. A. S., ”On n-Power Normal Operators” 푇∗is the dual operator of 푇. The Arabian Journal for Science and Engineering, Now the following problem can be raised: Can we establish similar results on *-푛-paranormal Vol. 33, No. 2A, pp. 247-253. operators in the Hilbert spaces and their tensor [13] KreyszigE.,”Introductory Functional Analysis product? with Applications”, Wiley Classics Library

References edition,1989. [1] Alzuraiqi S. A. & A. B. Patel, ”General [14] Limaye B.V., ”Functional Analysis”, New Age Mathematics Notes”, Vol.1 No.2, 2010, pp.61-73. International Publishers, Second Edition, 1995. [2] Braha N. L.&M.Lohaj& F.H. Marevci& [15] Panayappan S. & N. Sivamani, On n-Power Sh.Lohaj ”Some properties of paranormal and Class (Q) Operators, International Journal of hyponormaloperators”,Bulletin of Mathematical Mathematics, Vol.6, No. 31, 2010, pp. 1513-1518. analysis and Applications,Volume 1, Issue 2, 2009, [16] Stampfli J.G.,”Hyponormal and Spectral pp. 23-35 Density”,Pacific J.Math.16,1963, pp. 469-476. [3] Bartle R. G. & D. R. Sherbert, ”Introduction to [17] Stampfli J.G.,”HyponormalOperators ”Pacific J. Real Analysis”, Wiley India Edition, 1999. Math.12 1963, pp.1453-1458. [4] Bonsall F.F. &J.Duncan ”Complete Normed [18] Young M.H. &J.H.Son”On Quasi-M - Algebras”,SpringerVerlag,New York,1973. Hyponormal Operators”,Filomat 25:1,2011, pp.37- [5] Che-Kao F.”On M-hyponormal Operators”,Studia 52. Mathematica,T.LXV,1979. [19] Z. Lingling& A. Uchiyama ”On Polynomially *- [6] Cho M. & S. Ota ”On n-paranormal paranormal operators”,Functional Analysis, Operator”,Journal of Mathematics Research; Vol. 5, Analysis and Computation 5:2 2013, pp.11-16. No. 2, 2013.

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

Authors Profile P.J.Dowarireceived the M.S.degree in Mathematical Science from Gauhati University Guwahati, India, in 2015.Currently persuingPhD in Tripura University. His research interest includes Functional Analysis,Paste Operator Theory.

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here N.Goswamireceived the M.Scdegree in Mathematics from IIT, Delhi in 2004, and awarded Ph.D. Degree from Gauhati University, Assam, India. Currently working as an Assistant Paste Professor in the Dept. of Mathematics, Gauhati University,Photo Assam, India. Her research interest includes Functional Analysis, Topology, Fuzzy Set Theory.here