Open Mathematics 2021; 19: 32–45 Research Article Ahmed Bachir*, Abdelkader Segres, and Nawal A. Sayyaf Range-kernel weak orthogonality of some elementary operators https://doi.org/10.1515/math-2021-0001 received December 3, 2019; accepted October 12, 2020 Abstract: We study the range-kernel weak orthogonality of certain elementary operators induced by non- normal operators, with respect to usual operator norm and the Von Newmann-Schatten p-norm(1 ≤<∞p ). Keywords: range-kernel orthogonality, elementary operator, Schatten p-classes, quasinormal operator, sub- normal operator, k-quasihyponormal operator MSC 2020: 47A30, 47B20, 47B47, 47B10 1 Introduction Let B() be the algebra of all bounded linear operators acting on a complex separable Hilbert space . Given AB, ∈( B),wedefine the generalized derivation δAB, : BB()→() by δAB, ()=XAXX − B. ∗ 1 Let X ∈(B ) be a compact operator, and let s12()≥XsX ()≥…≥0 denote the eigenvalues of∣XXX∣=( )2 arranged in their decreasing order. The operator X is said to belong to the Schatten p-class p(),if 1 ∞ p pp1 ( ) ∥∥=XsXXp ∑ i () =tr (∣∣)<+∞p , 1 i=1 where tr denotes the trace functional. In case p =∞, we denote by ∞(), the ideal of compact operators equipped with the norm ∥∥XsX∞ =1 (). For p =(1, 1 ) is called the trace class, and for p =(2, 2 ) is called the Hilbert-Schmidt class and the case p =∞corresponds to the class. For more details, the reader is referred to [1]. In the sequel, we will use the following further notations and definitions. The closure of the range of an operatorTB∈() will be denoted by ran T and ker T denotes the kernel ofT. The restriction ofT to an invariant subspace will be denoted by T∣ , and the commutator AB− BA of the operators AB, will be denoted by []AB, . We recall the definition of Birkhoff-James’s orthogonality in Banach spaces [2,3]. Definition 1. If is a complex Banach space, then for any elements x, y ∈ , we say that x is orthogonal to y, noted by xy⊥ ,iff for all α, β ∈ there holds ∥+∥≥∥∥αy βx βx . (2) for all α, β ∈ or . * Corresponding author: Ahmed Bachir, Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia, e-mail: [email protected], [email protected] Abdelkader Segres: Department of Mathematics, Mascara University, Mascara, Algeria, e-mail: [email protected] Nawal A. Sayyaf: Department of Mathematics, College of Science, Bisha University, Bisha, Saudi Arabia, e-mail: [email protected] Open Access. © 2021 Ahmed Bachir et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. Range-kernel weak orthogonality 33 If and are linear subspaces in , we say that is orthogonal to if xy⊥BJ− for all xM∈ and all y ∈ N . The orthogonality in this sense is asymmetric. Let ∗→(: B ) be an involution defined on a linear subspace of B() onto the algebra of all bounded linear operators acting on the Banach space and ∗ = . According to the definition given by Harte [4], E is called the Fuglede operator if kerEE⊆ ker ∗. The elementary operator is an operator E defined on Banach () , -bimodule with its representa- n n n - tion Ex()=∑i=1 axbii, where a =(aii ) ∈ , b =(bii ) ∈ are n tuples of algebra elements. The length of E fi is de ned to be the smallest number of multiplication terms required for any representation ∑j axbj j for E. In this note, we consider ==( B ) and =(B ) or =(≤<∞p :1 p ) and the length of E will 2 be less or equal to 2, i.e., if AAABBB=(12,, ) =( 12 ,) are 2-tuples of operators in B() , then the elementary operator induced by A and B is defined by EX()= AXB11 − AXB 22for all X ∈ . We will denote by E˜ the ∗∗ fi ˜ ˜ ∗ ∗ formal adjoint of E de ned by EX()=∑iJ∈ Aii XB for all X ∈ .NotealsothatEE∣=22() ∣ and EX()= ∗ ∑iJ∈ BXAiion any separable ideal of compact operator, where E is the operator adjoint of E in the sense of duality. J. Anderson [5] proved that if A and B are normal operators, then for allXS ,∈() B : S ∈ ker δAB,, ⇒∥()+∥≥∥∥ δ AB X S S. (3) This means that the kernel of δA,B is orthogonal to its range. F. Kittaneh [6] extended this result to an u.i. ideal norm in B(), by proving that the range of δAB, ∣ is orthogonal to ker δAB, ∩ . A detailed study of range-kernel orthogonality for generalized derivation δA,B has received much atten- tion in recent years and has been carried out in a large number of studies [3,5,7–13] and are based on the following result. Theorem 2. ( ) Let AB, be operators in B( ). If δA,B is Fuglede, then the range of δA,B resp. the range of δAB, ∣p is ( ) orthogonal to the kernel of δA,B resp. the kernel of δAB, ∣p for all 1 ≤≤∞p . D. Keckic [14] and A. Turnšek [15] extended Theorem 2 to the elementary operator E defined by EX()= AXB − CXD, where ()AC, and ()BD, are 2-tuples of commuting normal operators. Duggal [16] generalized the famous theorem to the case ()AC, and ()BD, are 2-tuples of commuting operators, where AB, are normal and C, D∗ are hyponormal. In this paper, our goal is to extend the previous theorem to non-normal operators including quasi- normal, subnormal, and k-quasihyponormal operators. In the following, we recall some definitions about the range-kernel weak orthogonality. Definition 3. [4] If E : → and T : → are bounded linear operators between Banach spaces and 0 <≤k 1, [sT∈⇒()≥∥∥]⇔∠⇔∠ker dist sEsTET , rankk ker ran E. (4) We say that T is weakly orthogonal to E, written T ∠ E, or equivalently kerTEkkTE∠⇔∃<≤∠ ran : 0 1 : k . (5) For 0 <≤k 1, we say that ET, has a 1 -gap if TE. () k ∠k If T = E and k ≠ 1, we shall call E w-orthogonal EE, consequently we get a 1 -gap between the subspaces ( ∠ ) k ker E and ran E, which corresponds to the “range-kernel weak-orthogonality” for an operator E.Ifk = 1,we shall say that T is orthogonal to E, written T ⊥ E, also if =(Y )= and T = E we get a 1-gap between the subspaces ker E and ran E. 34 Ahmed Bachir et al. T is said to be a quasi-normal if [TTT, ∗ ]=0, subnormal if there exist a Hilbert space ⊇ and a normal operator N ∈(B ) such that NT∣= . Also, T is called hyponormal if []TT∗, is a positive operator. Furthermore, we have the following proper inclusion TTnormal⇒-⇒ quasi normal T subnormal ⇒ T hyponormal. AB∈() is said the Fuglede operator [13] if kerAA⊆ ker ∗. n Recall that an n-tuple AAAA=(12,,, …n )∈ B ( ) is said commuting (resp. doubly commuting) if ( ∗ ) - [AAij, ]=0 resp. [AAij, ]=0 and [AAi , j]=0 for all ij,1,,=… n, ij≠ . The n tuple A is said to be normal if A is commuting and each Aii (=1, … , n) is normal, and A is subnormal if A is the restriction of a normal n-tuple to a common invariant subspace. Clearly, every normal n-tuple is subnormal n-tuple. Any other notation or definition will be explained as and when required. 2 Preliminaries We summarize the results given by D. Keckic [14], A. Turnšek [15], and B. P. Duggal [16] in the following theorem. Theorem 4. [14–16] Let AB, be normal operators, C, D∗ be hyponormal operators in B() such that [AC,,]=[ BD ]=0 and =(B ) or =≤≤∞Cpp :1 , then (i) If kerAC∩={}=∩ ker 0 ker B∗∗ ker D, then for all X ∈(B ) such that EX˜(), EX ()∈ S ∈∩⇒{∥()+∥∥()+∥}≥∥∥kerEEXSEXSS min , ˜ . (ii) If kerAC∩≠{ ker 0} or kerBD∗∗∩≠{ ker 0}, then there exists k verifying 0 <<k 1 such that ∀X∈() B ,ker: ∀∈ S E ∩ ∥()+∥≥∥∥ EX S kS , whereEX ()∈ . ∗ (iii) If =(C2 ) with its inner product ⟨XY,tr⟩= ( YX) and E is defined on C2(), then for all SE∈ ker , we get 2 2 2 ˜˜2 2 2 ∥()+∥=∥()∥+∥∥EXSEXSEXSEXS2 2 2 ; ∥()+∥=∥()∥+∥∥2 2 2 . We recall some useful results which are important in the sequel. Lemma 5. [13] Let AB, be commuting operators in B() with no trivial kernel. 1. Let ξ be the elementary Fuglede operator defined by ξX()= AXA∗∗ − BXB, then (i) if kerAB∩={} ker 0 , then ker A reduces A and ker B reduces B. (ii) if [AB, ∗]=0 and kerAB∩≠{ ker 0}, then kerAB∩ ker reduces A and B. 2. Let AB∈() , BB ∈(), and EX()= AXBX;, ∈ B ( ), then E is the Fuglede operator if and only if ker A reduces A and ker B∗ reduces B∗. Lemma 6. [ ] n 9 Let T be an operator represented by block matrix as TT=(ij, )ij, =1. 1 22 2 (i) If TB∈(), then ∑∥TTij, ∥≤∥∥≤∑∥ Tij, ∥ n2 ij, ij, (ii) If TCT∈()≤<∞p ;1 p , then 1 ∥∥TTTforallpp ≤ ∥ ∥p ≤∥∥p 2, ≤ <∞ p−2 p ∑ ij, p p n ij, 1 ∥∥TTp ≤ ∥ ∥p ≤ ∥∥Tforallpp 12. ≤ ≤ p ∑ ij, p p−2 p ij, n Range-kernel weak orthogonality 35 3 Main results Proposition 7. Let , be Hilbert spaces and AB∈(), B ∈(B ), and EBB∈(( , )) such that EX()= AXB. If E is the Fuglede operator, then E is w-orthogonal, E ∠ E˜, and the inequality min{∥EX ( ) + S ∥ , ∥ EX˜ ( ) + S ∥ } ≥ ∥ S ∥ (6) holds if – =(B , ) with k =/12or 1 1 – C , with k , if 1 p 2 and k , if 2 p . =(p ) = 2−p ≤≤ = p−2 ≤<∞ 2 p 2 p Proof. If A and B∗ are injective operators, then E is injective. So there is nothing to prove. Suppose that A or B∗ is the non-injective operator.