EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 2073-2083 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some Properties of (M,k)−Quasi Paranormal Operators on Hilbert Spaces Valdete Rexhëbeqaj Hamiti1, Shkumbin Makolli2,∗ 1 Department of Mathematics, Faculty of Electrical and Computer Engineering, University of Prishtina ”Hasan Prishtina”, Prishtinë, 10000, Kosovë 2 Department of Mathematics, Faculty of Civil Engineering, University of Prishtina ”Hasan Prishtina”, Prishtinë, 10000, Kosovë Abstract. Let H be a complex Hilbert space and let T represent a bounded linear operator on H. In this paper we introduce, a new class of non normal operators, the (M,k)−quasi paranormal operator. An operator T is said to be a (M,k)−quasi paranormal operator, for a non negative integer k and a real positive number M if it satisfies ∥T k+1x∥2 ≤ M∥T k+2x∥·∥T kx∥, for all x ∈ H. This new class of operators is generalization of some of the non normal operators, such as, the k−quasi paranormal and M−paranormal operators. We prove the basic properties, the structural and spectral properties and also the matrix representation of this new class of operators. 2020 Mathematics Subject Classifications: 47B20, 47B47, 47A10 Key Words and Phrases: (M,k)−quasi paranormal operator, M−quasi paranormal operator, k−quasi paranormal operator, M− paranormal operator, approximate point spectrum of operator 1. Introduction Let H be a complex Hilbert space with inner product ⟨·, ·⟩. Let L(H) denote the C∗ algebra of all bounded operators on H. For an operator T ∈ L(H), by kerT and T (H) we denote the null space and the range of T, respectively. The null operator will be denoted by 0 and the identity operator by I. If T is an operator, then T ∗ is its adjoint, and ∥T∥ = ∥T ∗∥. By σ(T ), r(T ), σa(T ) we write the spectrum, the spectral radius and the approximate point spectrum of T, respectively. An operator T ∈ L(H) is said to be: an isometry if ∥Tx∥ = ∥x∥, for all x ∈ H; an unitary operator if T ∗T = TT ∗ = I and positive operator T ≥ 0, if ⟨Tx, x⟩ ≥ 0, for all x ∈ H (see [4], [8]). One of the attractive areas of research in operator theory is the study of non normal operators. Some of the interesting classes of non normal operators, which have been intro- duced and studied before are paranormal operator, M−paranormal operators, k−quasi ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5303 Email addresses: valdete.rexhebeqaj@uni-pr.edu (V. R. Hamiti), shkumbin.makolli@uni-pr.edu (Sh. Makolli) https://www.ejpam.com 2073 © 2024 EJPAM All rights reserved. V. R. Hamiti, Sh. Makolli / Eur. J. Pure Appl. Math, 17 (3) (2024), 2073-2083 2074 paranormal operators, M−quasi paranormal operators, ect. An operator T ∈ L(H) is said to be: a paranormal operator if ∥Tx∥2 ≤ ∥T 2x∥ for any unit vector x in H (see [2], [5], [6], [15]); a M−paranormal operators if ∥Tx∥2 ≤ M∥T 2x∥ for any unit vector x in H and for a fixed real positive number M (see [1], [3], [12]); a quasi paranormal operators if ∥T 2x∥2 ≤ ∥T 3x∥ · ∥Tx∥, for all x ∈ H (see [10]); a k−quasi paranormal operators if ∥T k+1x∥2 ≤ ∥T k+2x∥ · ∥T kx∥, for all x ∈ H and for a positive integer k (see [7], [13]); a M−quasi paranormal operators if ∥T 2x∥2 ≤ M∥T 3x∥ · ∥Tx∥, for all x ∈ H and for a fixed real positive number M (see [9]). In the present paper, we introduce a new class of operators (M,k)−quasi paranormal as a generalization of these non normal classes of operators. The purpose of this paper is, first to give some properties of this new class of operators, to compare this class with the other non normal classes of operators and also to study the structural and spectral properties of this class of operators. 2. Definition and Some Properties Definition 1. An operator T ∈ L(H) is said to be a (M,k)−quasi paranormal operator, for a non negative integer k and a real positive number M if it satisfies ∥T k+1x∥2 ≤ M∥T k+2x∥ · ∥T kx∥, for all x ∈ H. This definition is equivalent to T ∗k(M2T ∗2T 2 − 2λT ∗T + λ2)T k ≥ 0, for all λ > 0. Similarly as [9, Proposition 2]. Example 1. On the usual Hilbert space l2, let T be a weighted shift operator, defined by T (en) = |αn|en+1, where (en) is the standard basis and (αn) is a decreasing weighted sequence. Then, T is a (M,k)−quasi paranormal operator if and only if |αn+k| ≤ M |αn+k+1| for every n. Since T is a weighted shift, its adjoint T ∗ is also a weighted shift and we have: T ∗(en) = |αn−1|en−1, (T ∗T )(en) = α2 nen, (T ∗2T 2)(en) = α2 nα 2 n+1en, ... (T ∗(k+2)T k+2)(en) = α2 nα 2 n+1...α 2 n+kα 2 n+k+1en. Now, since T is a (M,k)−quasi paranormal operator then, M2T ∗(k+2)T (k+2) − 2λT ∗(k+1)T (k+1) + λ2T ∗kT k ≥ 0, for all λ > 0 ⇔ V. R. Hamiti, Sh. Makolli / Eur. J. Pure Appl. Math, 17 (3) (2024), 2073-2083 2075 M2α2 nα 2 n+1...α 2 n+kα 2 n+k+1 − 2λα2 nα 2 n+1...α 2 n+k + λ2α2 nα 2 n+1...α 2 n+k−1 ≥ 0, for all λ > 0 ⇔ α2 nα 2 n+1...α 2 n+k−1(M 2α2 n+kα 2 n+k+1 − 2λα2 n+k + λ2) ≥ 0, for all λ > 0 ⇔ M2α2 n+kα 2 n+k+1 − 2λα2 n+k + λ2 ≥ 0, for all λ > 0. By elementary properties of real quadratic forms, this gives 4α4 n+k − 4M2α2 n+kα 2 n+k+1 ≤ 0 |αn+k| ≤ M |αn+k+1|. From definition it is clear that this class of operators is nested with respect to M , i.e., a (M1, k)−quasi paranormal operator is (M2, k)−quasi paranormal operator for 0 < M1 < M2. From the above definition, the following facts follows: for M = 1, a (1, k)−quasi para- normal operator is a k−quasi paranormal operator; for k = 1, a (M, 1)−quasi paranormal operator is a M−quasi paranormal operator; for k = 0, a (M, 0)−quasi paranormal op- erator is a M−paranormal operator for any unit vector x in H; for M = 1, k = 0, a (1, 0)−quasi paranormal operator is a paranormal operator for any unit vector x in H; for M = 1, k = 1 a (1, 1)−quasi paranormal operator is a quasi paranormal operator. Important properties of this new class of operators are shown in the following theorems. Theorem 1. The class of (M,k)−quasi paranormal operators is closed under scalar mul- tiplication. Proof. Let T ∈ L(H) be a (M,k)−quasi paranormal operator, and let α be any com- plex scalar. For all x ∈ H we have ∥(αT )k+1x∥2 = |α|2k+2∥T k+1x∥2 ≤ |α|2k+2M(∥T k+2x∥ · ∥T kx∥) = M∥(αT )k+2x∥ · ∥(αT )kx∥. Then, αT is also (M,k)−quasi paranormal operator. Theorem 2. Let T ∈ L(H) be a (M,k)−quasi paranormal operator and let S ∈ L(H) be an isometric operator. If T double commutes with S, then TS is a (M,k)−quasi paranormal operator. Proof. Let T ∈ L(H) be a (M,k)−quasi paranormal operator. Let be S an isometric operator and let be B = TS. Since operator T double commutes with operator S we have TS = ST, S∗T = TS∗ and S∗S = I. Now, B∗k(M2B∗2B2 − 2λB∗B + λ2)Bk V. R. Hamiti, Sh. Makolli / Eur. J. Pure Appl. Math, 17 (3) (2024), 2073-2083 2076 = (TS)∗k(M2(TS)∗2(TS)2 − 2λ(TS)∗(TS) + λ2)(TS)k = S∗T ∗S∗T ∗...S∗T ∗(M2S∗T ∗S∗T ∗TSTS − 2λS∗T ∗TS + λ2)TSTS...TS = S∗kT ∗k(M2T ∗2T 2 − 2λT ∗T + λ2)T kSk = (TS)∗k(M2T ∗2T 2 − 2λT ∗T + λ2)(TS)k ≥ 0, for all λ > 0, so TS is a (M,k)−quasi paranormal operator. Theorem 3. Let T ∈ L(H) be a (M,k)−quasi paranormal operator. If S ∈ L(H) is unitarily equivalent to operator T, then S is a (M,k)−quasi paranormal operator. Proof. Let T ∈ L(H) be a (M,k)−quasi paranormal operator. Since operator S is unitarly equivalent to operator T, then there exists an unitary operator U such that S = U∗TU. Since T is a (M,k)−quasi paranormal operator then T ∗k(M2T ∗2T 2 − 2λT ∗T + λ2)T k ≥ 0. Hence, S∗k(M2S∗2S2 − 2λS∗S + λ2)Sk = (U∗TU)∗k(M2(U∗TU)∗2(U∗TU)2 − 2λ(U∗TU)∗(U∗TU) + λ2)(U∗TU)k = U∗kT ∗k(M2T ∗2T 2 − 2λT ∗T + λ2)T kUk ≥ 0, for all λ > 0, so S is a (M,k)−quasi paranormal operator. Theorem 4. Let T ∈ L(H) be a (M,k)−quasi paranormal operator. If A is a closed T invariant subset of H, then, the restriction T|A is a (M,k)−quasi paranormal operator. Proof. Let T ∈ L(H) be a (M,k)−quasi paranormal operator. ∥(T |A)k+1u∥2 = ∥T k+1u∥2 ≤ M(∥T k+2u∥ · ∥T ku∥) = M(∥(T |A)k+2u∥ · ∥(T |A)ku∥). This implies that T |A is a (M,k)−quasi paranormal operator. V. R. Hamiti, Sh. Makolli / Eur. J. Pure Appl. Math, 17 (3) (2024), 2073-2083 2077 Theorem 5. If T ∈ L(H) is a invertible (M,k)−quasi paranormal operator then T−1 is also (M,k)−quasi paranormal operator. Proof. Since T is a (M,k)−quasi paranormal operator, for a non negative integer k and a fixed real positive number M , we have ∥T k+1x∥2 ≤ M∥T k+2x∥ · ∥T kx∥, for all x ∈ H. Then, ∥T k+1x∥ ∥T k+2x∥ ≤ M∥T kx∥ ∥T k+1x∥ for each vector x ∈ H. Now replacing x by T−2k−2x, we have ∥T k+1T−2k−2x∥ ∥T k+2T−2k−2x∥ ≤ M∥T kT−2k−2x∥ ∥T k+1T−2k−2x∥ ∥T−k−1x∥ ∥T−kx∥ ≤ M∥T−k−2x∥ ∥T−k−1x∥ ∥T−(k+1)x∥2 ≤ M∥T−(k+2)x∥ · ∥T−kx∥ for each vector x ∈ H. This shows that T−1 is a (M,k)−quasi paranormal operator. Theorem 6. Let T ∈ L(H) be a (M,k)−quasi paranormal operator. If T k has dense range, then T is a M−paranormal operator. Proof. Let T ∈ L(H) be a (M,k)−quasi paranormal operator and let suppose that T k has dense range, T k(H) = H. Let x ∈ H, then there exists a sequence {xn}+∞ n=1 in H such that T k(xn) → x, n → +∞. Since T is a (M,k)−quasi paranormal operator, we have ⟨(M2T ∗(k+2)T (k+2) − 2λT ∗(k+1)T (k+1) + λ2T ∗kT k)xn|xn⟩ ≥ 0, for all λ > 0; ⟨(T ∗k(M2T ∗2T 2 − 2λT ∗T + λ2)T k)xn, xn⟩ ≥ 0, for all λ > 0; ⟨(M2T ∗2T 2 − 2λT ∗T + λ2)T kxn, T kxn⟩ ≥ 0, for all λ > 0. By the continuity of the inner product, we have ⟨(M2T ∗2T 2 − 2λT ∗T + λ2)x, x⟩ ≥ 0, for x ∈ H, for all λ > 0. Therefore T is a M−paranormal operator. V. R. Hamiti, Sh. Makolli / Eur. J. Pure Appl. Math, 17 (3) (2024), 2073-2083 2078 Theorem 7. Let T be a (M,k)−quasi paranormal operator. Then the tensor product T ⊗ I and I ⊗ T are both (M,k)−quasi paranormal operators. Proof. Since, T is (M,k)−quasi paranormal operator, then we have: T ∗k(M2T ∗2T 2 − 2λT ∗T + λ2)T k ≥ 0, for all λ > 0. Now, from the properties of tensor product (see [11], [14]) we have : (T ⊗ I)∗k[M2(T ⊗ I)∗2(T ⊗ I)2 − 2λ(T ⊗ I)∗(T ⊗ I) + λ2](T ⊗ I)k = (T ∗k ⊗ I)[M2(T ∗2T 2 ⊗ I)− 2λ(T ∗T ⊗ I) + λ2](T k ⊗ I) = [T ∗k(M2T ∗2T 2 − 2λT ∗T + λ2)T k]⊗ I ≥ 0. Therefore, T ⊗ I is (M,k)−quasi paranormal operator. Similarly, I ⊗ T is (M,k)−quasi paranormal operator. Theorem 8. If T ∈ L(H) is a regular (M,k)−quasi paranormal operator, then the ap- proximate point spectrum of operator T lies in the disc σa(T ) ⊆ {λ ∈ C : 1√ M∥T−k−1∥ · √ ∥T k+1∥ · ∥T k−1∥ ≤ |λ| ≤ ∥T∥}. Proof. Suppose that T is a regular (M,k)−quasi paranormal operator. For every unit vector x in Hilbert space H, we have: ∥x∥2 = ∥T−k−1 · T k+1x∥2 ≤∥T−k−1∥2 · ∥T k+1x∥2 ≤∥T−k−1∥2 ·M · ∥T k+2x∥ · ∥T kx∥ ≤M · ∥T−k−1∥2 · ∥T k+1∥ · ∥Tx∥ · ∥T k−1∥ · ∥Tx∥ =M · ∥T−k−1∥2 · ∥T k+1∥ · ∥T k−1∥ · ∥Tx∥2. So, 1 ≤ M · ∥T−k−1∥2 · ∥T k+1∥ · ∥T k−1∥ · ∥Tx∥2, where we have ∥Tx∥ ≥ 1√ M∥T−k−1∥ · √ ∥T k+1∥ · ∥T k−1∥ . Now, assume that λ ∈ σa(T ), then there exists a sequence (xn), such as ∥xn∥ = 1 and ∥(T − λI)xn∥ → 0, n → +∞. V. R. Hamiti, Sh. Makolli / Eur. J. Pure Appl. Math, 17 (3) (2024), 2073-2083 2079 From the last inequation we have: ∥Txn − λxn∥ ≥ ∥Txn∥ − |λ| · ∥xn∥ ≥ 1√ M∥T−k−1∥ · √ ∥T k+1∥ · ∥T k−1∥ − |λ|. Now, when n → +∞ we have |λ| ≥ 1√ M∥T−k−1∥ · √ ∥T k+1∥ · ∥T k−1∥ . So, we have σa(T ) ⊆ {λ ∈ C : 1√ M∥T−k−1∥ · √ ∥T k+1∥ · ∥T k−1∥ ≤ |λ| ≤ ∥T∥}. 3. Matrix Representation of (M,k)−quasi paranormal operators In this section we represent some results for the matrix representation of (M,k)−quasi paranormal operators. Theorem 9. Let T ∈ L(H⊕H) be the operator defined as T = ( A B 0 0 ) . If A is a M−paranormal operator, then T is a (M,k)−quasi paranormal operator. Proof. Let D = M2A∗2A2 − 2λA∗A+ λ2. Similarly as [9, Proposition 9] we have: T ∗ = ( A∗ 0 B∗ 0 ) , T ∗(k+2) = ( A∗(k+2) 0 B∗A∗(k+1) 0 ) , T (k+2) = ( A(k+2) A(k+1)B 0 0 ) , T ∗(k+2)T (k+2) = ( A∗(k+2)A(k+2) A∗(k+2)A(k+1)B B∗A∗(k+1)A(k+2) B∗A∗(k+1)A(k+1)B ) . After some calculations, we have: T ∗k(M2T ∗2T 2 − 2λT ∗T + λ2)T k =M2T ∗(k+2)T (k+2) − 2λT ∗(k+1)T (k+1) + λ2T ∗kT k = ( A∗kDAk A∗kDA(k−1)B B∗A∗(k−1)DAk B∗A∗(k−1)DA(k−1)B ) V. R. Hamiti, Sh. Makolli / Eur. J. Pure Appl. Math, 17 (3) (2024), 2073-2083 2080 Let u = x⊕ y ∈ H ⊕H. Then, ⟨(T ∗(k+2)T (k+2) − 2λT ∗(k+1)T (k+1) + λ2T ∗kT k)u, u⟩ = ⟨A∗kDAkx, x⟩+ ⟨A∗kDA(k−1)By, x⟩ + ⟨B∗A∗(k−1)DAkx, y⟩+ ⟨B∗A∗(k−1)DA(k−1)By, y⟩ = ⟨DAkx,Akx⟩+ ⟨DA(k−1)By,Akx⟩ + ⟨DAkx,A(k−1)By⟩+ ⟨DA(k−1)By,A(k−1)By⟩ = ⟨D(Akx+A(k−1)By), (Akx+A(k−1)By)⟩ ≥ 0 because A is a M−paranormal operator then, D = A∗2A2 − 2A∗A+ I ≥ 0, so this proves the result. Theorem 10. If T k does not have a dense range, then the following statements are equiv- alent: (i) Operator T is a (M,k)−quasi paranormal operator, for a non negative integer k; (ii) T = ( A B 0 C ) on H = T k(H) ⊕ kerT ∗k, where A is a M−paranormal operator on T k(H), Ck = 0 and σ(T ) = σ(A) ∪ {0}. Proof. (1) ⇒ (2) Similarly as [9, Proposition 10]. (2) ⇒ (1) Suppose that T = ( A B 0 C ) on H = T k(H) ⊕ kerT ∗k, where A is a M−paranormal operator on T k(H), and Ck = 0. A simple calculation shows that: T ∗ = ( A∗ 0 B∗ C∗ ) , T ∗T = ( A∗A A∗B B∗A B∗B + C∗C ) , T ∗2T 2 = ( A∗2A2 A∗2AB +A∗2BC B∗A∗A2 + C∗B∗A2 |AB +BC|2 + |C2|2 ) , T ∗k = ( A∗k 0 ( ∑k−1 j=0 A jBCk−1−j)∗ 0 ) , T k = ( Ak ( ∑k−1 j=0 A jBCk−1−j) 0 0 ) , Then, we have T ∗k(M2T ∗2T 2 − 2λT ∗T + λ2)T k V. R. Hamiti, Sh. Makolli / Eur. J. Pure Appl. Math, 17 (3) (2024), 2073-2083 2081 = ( A∗k 0 ( ∑k−1 j=0 A jBCk−1−j)∗ 0 ) × ( D A∗2AB +A∗2BC B∗A∗A2 + C∗B∗A2 − 2λB∗A |AB +BC|2 + |C2|2 − 2λ(B∗B + C∗C) + λ2 ) × ( Ak ∑k−1 j=0 A jBCk−1−j 0 0 ) = ( A∗kDAk A∗kD ∑k−1 j=0 A jBCk−1−j ( ∑k−1 j=0 A jBCk−1−j)∗DAk ( ∑k−1 j=0 A jBCk−1−j)∗D ∑k−1 j=0 A jBCk−1−j ) , where D = M2A∗2A2−2λA∗A+λ2. Let v = x⊕y be a vector in H = T k(H)⊕kerT ∗k, where x ∈ T k(H) and y ∈ kerT ∗k. Then, 〈 T ∗k(M2T ∗2T 2 − 2λT ∗T + λ2)T kv, v 〉 = 〈 A∗kDAkx, x 〉 + 〈 A∗kD k−1∑ j=0 AjBCk−1−jy, x 〉 + 〈 ( k−1∑ j=0 AjBCk−1−j)∗DAkx, y 〉 + 〈 ( k−1∑ j=0 AjBCk−1−j)∗D k−1∑ j=0 AjBCk−1−jy, y 〉 = 〈 D(Akx+ k−1∑ j=0 AjBCk−1−jy), Akx+ k−1∑ j=0 AjBCk−1−jy 〉 . Since A is a M−paranormal operator we have that D = M2A∗2A2 − 2A∗A + I ≥ 0. Therefore, 〈 T ∗k(M2T ∗2T 2 − 2T ∗T + I)T kv, v 〉 ≥ 0 for all v ∈ H. Hence, T ∗k(M2T ∗2T 2 − 2T ∗T + I)T k ≥ 0 So we have that T is a (M,k)−quasi paranormal operator. 4. Conclusion In this paper we have introduced a new class of operators in Hilbert spaces, which we named the (M,k)−quasi paranormal operators. First we have proved some basic properties REFERENCES 2082 and also the structural and spectral properties of this class of operators. We also have given the relations of with new class of operators with other non normal classes of operators in Hilbert spaces. We also have given an example that support the theoretical approach. 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