Range-Kernel Weak Orthogonality of Some Elementary Operators

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, subnormal 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]

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 attention 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 quasinormal, 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 <

∀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. From Lemma 5(2) and with respect to the decompositions: =(kerAA )⊥∗⊥∗ ⊕ ker , =( ker B ) ⊕ ker B we obtain AA=⊕1 A; B =⊕B1 0. Let X =(Xij,,1,2 ) ij= : → . Then

EX()= AX1111 B ⊕0.

∗ From the injectivity of A1 and B1 it yields that any S =(Sij ) i,1, j= 2 in ker E can be written as

 0 S12  S =  , SS21 22 where the operators S12, S 21, and S22 are arbitrary. Hence, for all SE∈ ker and all X ∈(B , ), by Lemma 5(2), we have

AX B S 1 1 ˜  1111 12 2 2 212/ min{∥EX ( ) + S ∥ , ∥ EX ( ) + S ∥} =  ≥ (∥SSS12 ∥ + ∥21 ∥ + ∥22 ∥ ) ≥ ∥ S ∥.  SS21 22 2 2

Also, for all S ∈∩(kerECp , ) and all X ∈(B , ) such that EX()∈ Cp ( , ), we get (i) if 1 ≤≤p 2, then

AX B S  1 1 min{∥EX ( ) + S ∥ , ∥ EX˜ ( ) + S ∥ } =1111 12 ≥ (∥SSS ∥p + ∥ ∥p + ∥ ∥p )1/p ≥ ∥ S ∥ . pp SS 22−−p 12p 21p 22 p p p  21 22 p 2 p 2 p

(ii) if 2 ≤<∞p , then

AX B S  1 1 min{∥EX ( ) + S ∥ , ∥ EX˜ ( ) + S ∥ } =1111 12 ≥ (∥SSS ∥p + ∥ ∥p + ∥ ∥p )1/p ≥ ∥ S ∥ . pp SS p−−2212p 21p 22 p p p  21 22 p 2 p 2 p □ Using Lemma 5(2), we get a simple form of the previous result as follows.

Corollary 8. If ker A reduces A and ker B∗ reduces B∗, then E is w-orthogonal, E ⟨E˜ and satisﬁes the relation (6).

In the sequel ξ denotes the elementary operator deﬁned by ξX()= AXA∗∗ − BXB from B() to , where A and B are operators in B() and =(B ) or =(Cp ); 1 ≤<∞p .

Lemma 9. Let Δ be the elementary operator deﬁned on B() by Δ()=XAXB −X, where AB, ∈( B). If Δ is a Fuglede operator, then Δ is orthogonal and ΔΔ⊥ ˜ .

Proof. The proof is the same as the one in Theorem 4. □ 36  Ahmed Bachir et al.

Proposition 10. Let AB, be doubly commuting operators in B() and

ξB:;1,2()→() Cp (≤≤∞ p p ≠).

If ker A reduces A, ker B reduces B and ξ orthogonal, then kerAB∩={ ker 0}.

Proof. We consider the following three cases: (i) Let us suppose that =∩≠{kerAB ker 0}. If kerAB≠ ker and kerBA⊈ ker , then with respect to the decomposition =(kerBB )⊥ ⊕( ker ⊖ )⊕ and from the hypothesis it yields

AA=⊕⊕12 A0, BB =⊕ 1 0, and S =() Sij i ,1,,3 j=… ,

where A2 is an injective operator. Hence, ∗∗∗∗ S ∈⇒kerξASABSBASAASA1111 =111 1 ;112 2 =2211 = 0; S22 =0 and the other entries are arbitrary. Choosing X and S as follows:

0 R X =⊕(⊗)⊕00,0ee S =⊕ , RC∗  where e is a non-zero vector in , R is an operator of rank one, and C is a self-adjoint operator of rank one. Then

Ae⊕ Ae R ξX()+ S =0 ⊕ 22.  RC∗  Applying Lemma (2.4)[15], we get

0 R ∥(ξX )+ S ∥ <  =∥∥Sp ( ≠2 ). p  ∗  p RC p

(ii) If kerAB≠ ker and kerAB⊈ ker , we proceed similarly as in the ﬁrst case, it suﬃces to replace A by B and B by A in the preceding argument. (iii) If kerAB= ker , then with respect to the decomposition =(kerBB )⊥ ⊕ ker , we get

AA=⊕110, BB =⊕ 0.

Letting S =(Sij ) i,1,,3 j=… . Then ∗∗ S ∈⇒kerξASABSB1111 =111 1 , and the other entries are arbitrary. Choosing X and S as

0 R X =(ee ⊗ )⊕0, S = , RC∗  where e and R are as in (i). Then

Ae⊗−⊗ Ae Be Be R ξX()+ S = 1111.  RC∗ 

If Ae1111⊗=⊗ Ae Be Be for all e ∈ , then it follows from this fact and the injectivity of A1 and B1 that AαB1 = 1 with ∣α∣=1 and hence ξ = 0, which is a contradiction with the assumption ξ ≠ 0. We use Lemma (2.4)[15] to complete the proof for p ∉{1, 2}. Range-kernel weak orthogonality  37

In the case p = 1, let us assume that Ae1111⊗−⊗ Ae Be Be is an operator of rank 2 and has eigenvalues λλ12, with ∣λλ12∣=∣ ∣ for all e ∈ , then we can check that

∣λ12∣=∣ λ ∣⇔∥ Ae 1 ∥=∥ Be 1 ∥and ⟨ Ae 11 , Be ⟩= 0 ∀ e ∈.

∗ ∗ If kerBB⊆ ker , then by the injectivity of B1 , it follows that A1 = 0, which is a contradiction since A ≠ 0. If kerBB∗ ⊈ ker , with respect to the decomposition ∗ ⊥ ∗ =(kerBBB11 ) ⊕ ker ⊕ ker we get

BB12 AA=⊕⊕12 A0, B = ⊕0, and SS =()ij i,1,,3 j=… .  00

Since A1 is injective and S ∈ ker ξ , we obtain ∗∗ AS111 A1 ==== BS111 B1 ; S12 S 21 S 22 0 and the other entries are arbitrary. We rewrite S on the following decomposition ∗ ⊥ ∗ =(kerBBB11 ) ⊕ ker ⊕ ker

∗ and choose S11===SS 23 32 0, SR13 = , S31 = R , SC34 = , and X =⊗⊕ee0 (RCe,, are as in (i)). Then

Ae⊗−⊗ Ae Be Be R ξX()+ S = 1111 ⊕ 0.  RC∗ 

If Ae1111⊗−⊗ Ae Be Be is an operator of rank two and has eigenvalues λλ12, with∣λλ12∣=∣ ∣ for all e ∈ , then from the previous argument we conclude that A1 = 0. On the other hand, if B2 ≠ 0,thenfrom [AB, ]=0 it follows that A2 = 0, also a contradiction with the fact that A ≠ 0. ∗ ∗ If B2 = 0 (ker B reduces B ), then B =⊕B1 0, AA=⊕0 2 ⊕0 (A2 is injective), and S =(Sij ) i,1,,3 j=… . Hence, it follows from ASA∗∗= BSB that

 0 SS12 13 S = SS0   21 23 SSS31 32 33 and the other entries are arbitrary. To conclude the proof we can argue similarly as in the ﬁrst case (i). □

Corollary 11. Let (AA12,,,)( BB 12) be 2-tuples of doubly commuting operators in B() and EB: ()→( Cp ); (1,2≤≤∞≠pp) be the elementary operator deﬁned by EX()= AXB11 − AXB 22such that kerAA12 , ker , ∗ ∗ ∗ ∗ ker B1 , and ker B2 reduce AAB12,,1 , and B2 , respectively. If E is orthogonal, then ∗∗ kerAA12∩={}=∩ ker 0 ker B12 ker B.

Proof. Consider the space ⊕ and the following decompositions

∗∗0 X AA=⊕1 B,,and BA =⊕2 B Y = . 1 2 00

∗∗ ∗ Then, for all X ∈(B ),wehaveAYA−= BYB A11 XB − A 2 XB2, kerAA=∩ ker1 ker B1 ,andkerBA= ker 2 ∩ ∗ ker B2 .Hence, ∗∗ kerAA12∩={}=∩⇔∩={} ker 0 ker B12 ker B ker AB ker 0 . So to achieve the proof, it suﬃces to apply the previous proposition. □ 38  Ahmed Bachir et al.

Theorem 12. Let AAN121,,, and N2 be operators in B() such that ()NN12, is 2-tuple normal and [AN11, ]=

[AN22, ]=0. Let EX()= AXA12 − NXN 12 such that EX(), EX˜ ()∈ . If E is the Fuglede operator, then E is w-orthogonal and E ∠ E˜. Furthermore, (i) If =()(≤<∞≠Cppp :1 , 2), then

∗ ∗ E is orthogonal if and only if kerAN11∩={}= ker 0 ker A2 ∩ ker N2 ; ( ) ∗∗ ˜ ii If kerAN11∩={}=∩ ker 0 ker A22 ker N, then E is orthogonal and E ⊥ E; ( ) ∗∗ iii If kerAN11∩≠{ ker 0} or kerAN22∩≠ ker 0, then for all X ∈(B ) and all S ∈∩ker E ,

min{∥EX ( ) + S ∥ , ∥ EX˜ ( ) + S ∥ } ≥ kS ∥ ∥ ,

where 1 – =()Bk : = ; 2n 1 1 – Ck : , if 1 p 2 and k , if 2 p . =()p =42− p ≤≤ = 24p− ≤<∞ 2 p 2 p Proof. (i) The implication (⇒) follows from Corollary 8. Let ξB: ()→ be the elementary operator deﬁned by ξX()= AXA∗∗ − NXN, where AN, ∈() B , N is normal with [AN, ]=0. Assume that ξ is the Fuglede operator. ( ) −1 ii Suppose that N is invertible and set DN= A. ξ is Fuglede implies that ξD is Fuglede, where ξD is the elementary operator induced by D. By Lemma 2.4 [16], we have

∥()+∥=∥ξ X S AXA∗∗ − NXN +∥=∥( S D NXN ∗∗∗ ) D − NXN +∥≥∥∥ S S .

Similarly, it can be shown that ∥(ξX˜ )+ S ∥ ≥∥∥ S for any operator S ∈(ker ξ )∩ . (iii) Suppose that N is injective, set Δ:;λλn1 and μ Δ denotes the corresponding n ={ ∈ ∣ ∣≤n ∈ } ( n) spectral projection, where Pnn=−(IμΔ ) converges strongly to I. ∗ From [AN, ]=0 and by Fuglede-Putnam’s theorem it follows that [AN,0]= and therefore Pn reduces both A and N. Let

=⊕(−)PIPnn,

then AA=⊕12nn A; N =⊕NN12nnand Pn =⊕I 0, where N1n is invertible. Hence, ∗∗ Pnnn((ξX )+ SP ) =[ A111 XA1n − N111n XN1n + S11 ]⊕0 for all X =(XBij ) i,1,2 j= ∈ ( ) and S =(Sξij ) i,1,2 j= ∈ker implying ∗∗ ∥()+∥≥∥(()+)∥=∣ξX S PξXnnn SP A111 XA1n − N111n XN1n + S11 ∥≥∥∣ S 11 .

∗ ∗∗ Since Pnn(()+)ξX SP =[ AXA1n 11 1n − NXN1n 11 1n + S 11 ]⊕0, then ∗∗ ∗∗ ∥()+∥≥∥(()+)∥=∥ξX S PξXnn SP AXA1n 11 1n − NXN1n 11 1n + S 11 ∥≥∥∥ S 11 . On the other hand, for all positive integer n, we have

∗ ∥SPSPξXSξXS11 ∥ =∥nn ∥≤min {∥( )+ ∥ , ∥ ( )+ ∥ }<∞

and thus sup ∥∥<∞PSPnn for all S ∈∩ker ξ and X ∈(B ). It follows by Lemma 3 [14] that n min{∥ξX ( ) + S ∥ , ∥ ξX˜ ( ) + S ∥ } ≥ ∥ S ∥ .

(iii) Suppose that N is an arbitrary normal operator. If kerAN∩={ ker 0}, then may be decomposed as =[(kerNAAN )⊥ ⊖ ker ]⊕ ker ⊕ ker . Range-kernel weak orthogonality  39

Since ξ isFuglede and by Lemma 5(1.(i)),wehaveker A reduces A, AA=⊕⊕110 A 22 and N = N11⊕⊕N 22 0. ∗∗ For X =(XBij ) i, j= 1,2,3 ∈ ( ), set ξX1()=11 AXA 11 11 11 − NXN11 11 11 and S =(Sξij ) i, j= 1,2,3 ∈ker , then ∗∗∗ ξS1()=110and ASA 11 13 22 = ASA22 31 11 = AS22 33 A22 = 0, ∗∗∗ N11SN 12 22 == NSN22 21 11 NSN22 22 22 =0. Since ξ is Fuglede, then S ∈ ker ξ˜ and ∗∗∗∗ ( ) ξS1 ()=11 0and ASA11 13 22 = ASA22 31 11 = ASA22 33 22 = 0, 7 ∗∗∗ N11SN12 22 == NSN22 21 11 NSN22 22 22 =0. ( ) ˜ Since N11,,NA 22 11, and A22 are injective, we get from ii that ξ1 is orthogonal, ξξ1 ⊥ 1 and S13===SS 31 33 S12===SS 22 21 0 and any operator S ∈ ker ξ has the form

S11 00 S =  00S ,  23  00S32  where S23 and S32 are arbitrary. Let S23=∣∣=∣US 23 23, S 32 US 32 32∣ be the polar decompositions of S23 and S32, respectively, let V be the operator deﬁned by

∗  0 U32 VI=⊕ ∗ . U23 0  Then

ξX1()+11 S 11 ∗ ∗   ξX S VξX S . ∥( )+ ∥ ≥∥(( )+ )∥=  ∗∣∣∗S32   ∗∗∣∣S23  Applying Lemma 6, we get

– if =(B ):

∥ξX ( ) + S ∥ ≥max {∥ ξ1 ( X11 ) + S 11 ∥ , ∥ S 32 ∥ , ∥ S 23 ∥} ≥ max {∥ S 11 ∥ , ∥ S 32 ∥ , ∥ S 23 ∥} = ∥ S ∥.

– if =(Cp ); (1 ≤<∞p ):

p p p 1//p p p p 1 p ∥ξX ( )+ S ∥p ≥(∥( ξ1 X11 )+ S 11 ∥p +∥ S 32 ∥p +∥ S 23 ∥p ) ≥(∥ S11 ∥p +∥ S 32 ∥p +∥ S 23 ∥p ) =∥ S ∣p . By the same method, we have that ξξ⊥ ˜. If M =∩≠{kerAN ker 0} and is decomposed as =(kerNNMM )⊥ ⊕[ ker ⊖ ]⊕ , then by Lemma 5(ii) and the fact that ξ is Fuglede, it follows that M reduces A, and hence

AA=⊕⊕11 A 220, NN =⊕ 11 0.

∗∗ For X =(Xij ) i,1,2,3 j= , we set ξX1()= AXA11 11 11 − NXN11 11 11 and let S =(Sξij ) i, j= 1,2,3 ∈ker . From the injectivity of A11 and A22, we obtain

ξS1()=110and S 12 = S 21 = S 22 = 0. By simple computation, we get

ξX1()+11 S 11 ∗ S 13   ξX S . ∥( )+ ∥ =  ∗∗S23  SSS31 32 33 40  Ahmed Bachir et al.

Applying Lemma 6 yields

– for =(B ):

2 1 2 2 2 2 2 2 1 2 ∥()+∥≥ξX S (∥( ξ X11 + S 11 )+∥ S13 ∥+∥ S23 ∥+∥ S32 ∥+∥ S13 ∥+∥ S33 ∥)≥ ∥∥ S . 22 1 24

– for =(Cp ); 1 ≤≤p 2:

p 1 p p p p p p 1 p ∥()+∥≥ξX S (∥( ξ X11 + S 11 )+∥∥+∥∥+∥∥+∥∥+∥∥)≥ S 13 S 23 S 32 S 13 S 33 ∥∥ S . p 22−p 1 p p p p p p 242− p p

– for =(Cp ); 2 ≤<∞p :

p 1 p p p p p p 1 p ∥()+∥≥ξX S (∥( ξ X11 + S 11 )+∥∥+∥∥+∥∥+∥∥+∥∥)≥ S 13 S 23 S 32 S 13 S 33 ∥∥ S . p 22−p 1 p p p p p p 224p− p Similarly,

– for =(B ): 1 ∥(ξX˜ )+ S ∥≥ ∥∥ S. 22

– for =(Cp ); 1 ≤≤p 2: ˜ 1 ∥(ξX )+ S ∥pp ≥42− p ∥∥ S . 2 p

– for =(Cp ); 2 ≤<∞p : ˜ 1 ∥(ξX )+ S ∥pp ≥24p− ∥∥ S . 2 p

Let us now ﬁnish the proof for the elementary operator EX()= AXA12 − NXN 12. Consider the space ⊕ and deﬁne the following operators on B(⊕) as

∗∗ 0 X N =⊕NN1 ;,and AAA =⊕1 Z = . 2 2 00

∗∗ It is clear that N is normal operator and [AN11,,]=[ AN 22 ]=0 imply [AN, ]=0 and EZ()= AZA − NZN. Applying the preceding result, the proof is complete. □

Let AB, ∈( B) and Ω be a set in B() 2 defined by (AB,Ω)∈ if and only if Δ()=XANXBMX − is the Fuglede operator for any normal N, M satisfying [NA,,]=[ MB ]=0. It follows from the definition that Δ is Fuglede and Ω ≠∅since (II,Ω)∈ . It is shown in [13] that if (AB,Ω)∈ , then the elementary operator E defined by EX()= NXM − AXBis Fuglede for any normal operators N and M in B(). So as a consequence of the previous theorem, we get the following corollaries.

Corollary 13. Let AB, ∈( B) and N, M be arbitrary normal operators in B() such that[NA,,]=[ MB ]=0 and E be the elementary operator deﬁned by EX()= NXM − AXBfor all X ∈(B ). If (AB,Ω)∈ , then E is w-orthogonal and E ∠ E˜. Furthermore, E and E˜ verify assertions (i), (ii), and (iii) in Theorem 12.

Lemma 14. [13] Let AB, ∈( B) and N, M be normal operators in B() such that [NA,,]=[ MB ]=0, then (AB,Ω)∈ in each of the following cases: (i) A and B∗ are hyponormal operators; (ii) A is k-quasihyponormal and B∗ is injective k-quasihyponormal operator. Range-kernel weak orthogonality  41

Corollary 15. If (AA12,,,)( BB 12) are 2-tuples of commuting operators in B() such that AB1, 1 are normal operators and EX()= AXB11 − AXB 22, then E is w-orthogonal and E ∠ E˜. Furthermore, E and E˜ satisfy assertions (i), (ii), and (iii) cited in Theorem 12, in each of the following cases: ( ) ∗ i A1 and B1 are hyponormal operators; ( ) - ∗ - ii A1 is k quasihyponormal and B1 is injective k quasihyponormal operator.

In the next theorem, we give a positive answer to a question raised by P. B. Duggal [16]: Is Theorem 2 still true if the hypothesis is related to A and B∗ being subnormal?

∗ 2 Theorem 16. If A and B are 2-tuples of commuting subnormal operators in [B()] such that AAA=( 12, ),

B =(BB12, ), and EX()= AXB11 − AXB 22, then E is w-orthogonal and E ∠ E˜.

Proof. From the deﬁnition of subnormality of 2-tuple operator, we have A is the restriction of a 2-tuple ∗ normal N =(NN12, ) to a common invariant subspace and B is the restriction of a 2-tuple normal ∗ M =(MM12, ) to a common invariant subspace equivalently to ANii=∣; Bi = Mi ∣, i = 1, 2 with Ni, Mi are normal operators on a Hilbert space ⊇ and [NN12,,]=[ MM 1 2 ]=0.IfSE∈ ker , then for all X ∈(B ), we have

˜˜∗∗˜ EX()+ S 0 N1XM1 −+= N2 XM2 S ,  00 where X˜ =⊕X 0 and S˜ =⊕S 0. Hence, ˜˜∗∗˜ ∥−NXM1 1 NXM2 2 +∥=∥()+∥ S EX S .

Since Nii,:Mi= 1,2 are normal, we get the w-orthogonality of E. With similar argument, E ∠ E˜ follows. □

Corollary 17. - 2 ∗ If ()AA12, and B =(BB12, ) are 2 tuples of commuting operators in [B()]such that [AAAi , 1 1 ∗∗∗∗ - ˜ + AA2 21]=[ Bi ,0;1,2 BB1 + BB2 2 ]= i = and EX()= AXB11 − AXB 22, then E is w orthogonal and E ∠ E.

Proof. By assumption, A and B∗ are spherically quasi-normal commuting 2-tuples (see deﬁnition [17]) and also by [17], A and B∗ are subnormal 2-tuples. So the desired result follows from Theorem 16. □

Theorem 18. Let (AA12,,,)( BB 12) be 2-tuples of doubly commuting operators in B() and E be the elementary ˜ ∗ operator deﬁned by EX()= AXB11 − AXB 22such that EX(), EX ()∈ ,, A1 B1 are quasi-normal operators and ∗ - ∗ ∗ - - AB2 , 2 are k quasihyponormal operators (k ≥ 1) with kerAA2 ⊆ ker 2 and kerBB2 ⊆ ker 2. Then E is w ortho gonal and E ∠ E˜. Furthermore, 1. If =(Cp ); (1,2≤<∞≠pp), then ∗∗ E is orthogonal if and only ifker A11∩= ker N ker A22 ∩ ker N ={} 0 ; ∗∗ ˜ 2. If kerAN11 ker={ 0 }= ker A22 ∩ ker N, then E is orthogonal and E ⊥ E; ∗∗ 3. If kerAA12∩≠ ker 0 or kerBB12∩≠{ ker 0}, then for all X ∈(B ) and all S ∈∩ker E ,

min{∥EX ( ) + S ∥ , ∥ EX˜ ( ) + S ∥ } ≥ kS ∥ ∥ . 1 – For =(B ), k = ; 6 1 – For C , k , if 1 p 2; =(p ) = 2−p ≤≤ 6 p 1 – For C , k , if 2 p . =(p ) = p−2 ≤<∞ 6 p 42  Ahmed Bachir et al.

Proof. Consider the following decompositions ⊥ ∗ ⊥ ∗ ==()⊕12kerAA ker 2 , ==()⊕ 2 ker BB22 ker . Then

AC21=⊕0, BC 22 =⊕ 0, ∗ - where C1, C2 are injective k quasihyponormal and by hypothesis, we get

ATT112=⊕, BRR 11 =⊕2 ∗ ∗ - with TTR12,,1 , and R2 are quasinormal operators and (TC11,,)( RC 12 ,) are 2 tuples of doubly commuting operators. ( ∗ ∗) Since ker T1 reduces T1 resp. ker R1 reduces R1 and by commutativity, we have that

TA1111=⊕0, RB =⊕ 11121222212 0, CA =⊕ A , CB =⊕ B2 with respect to the following decompositions

⊥⊥ ∗ ⊥ ∗ ⊥ ∗ (kerATTBRR211)=( ker )⊕ ker , ( ker211 ) =( ker )⊕ ker , ∗ ∗ ∗ - - where A11 and B11 are injective quasinormal operators, AAB21,, 22 21, and B22 are injective k quasihypo ∗ ∗ normal. We set AUA11=∣ 11∣ and B11=∣VB 11 ∣=∣∣ B11 V. Then it follows from the injectivity of A11 and B11 ∗ ∗ that U and V are isometry operators and by [TC11,,1]=[ T1 C ]=0, we get that ∗ ( ) [AA11,,0 21]=[ AA11 21 ]= . 8

∗∗ ∗ ∗ Then [AA11 11,, A 21]=[ AA 11 11 A21 ]=0 and [∣∣AA11,, 21 ]=[∣∣ AA11 21 ]=0.Hence,[UA,,21]=[ U A 21 ]=0. ∗ ∗ ∗ - Similarly, we obtain that[VB,,021]=[ V B 21 ]= , and (UA21) is k quasihyponormal. Indeed,

∗ ∗∗k ∗∗k kkkk+∗1 ∗ ∗+k 1 ∗k kk∗∗∗ k ()[AU21 UA21,,, AUA21 ] 21 U =()[ A21 U U UUA ]21 + UUAA21 [21 A21 ] A 21 UU ∗∗k kk∗∗∗ k =[]≤UU A21 A21,0. A21 A 21 UU

∗∗ Similarly, we get (VB21) is an injective k-quasihyponormal. Let X,:SB∈() X =() Xij i,1,2,3 j= and S =(Sij ) i, j= 1,2,3.IfSE∈ ker , then

ASB11 11 11= ASB 21 11 21, (9)

AS11 13 R 2=== TSB 2 31 11 TSR 2 33 2 0, (10)

ASB21 12 22=== ASB 22 21 21 ASB 22 22 22 0. (11) And

∗∗∗ ∗∗∗∗21 U (−AXB11 11 11 AXB 21 11 21 +)=∣∣∣∣−+ S 11 V A 11 X 11 B11 UAXBV21 11 USV11 . We derive from 8 that

∗∗∗∗∗ ∣A11∣∣∣= USVB 11 11 UAUSVBV 21 11 21 . Applying Corollary 8,

∗∗∗∗ ∥−AXB11 11 11 AXB 21 11 21 +∥≥∥(− S 11 UAXB 11 11 11 AXB 21 11 21 +)∥≥∥∥ S 11 V USV 11 . from the injectivity of A11 and its polar decomposition, we have

⊥⊥⊥ (kerTA11111)=( ker )=( ker UA ) ; ker = ran U and U :ker()⟶U ⊥ ranU is unitary. Taking the following decompositions yields (kerUUUV)=⊥⊥∗⊥∗∗⊥ ran ⊕( ran ) ; ( ker )= ran VV ⊕( ran ). Range-kernel weak orthogonality  43

Then

1 1 1 2 Aγ B 0 SS11 11 A 11 , B 11 ,,S 11 ===  11   11  3 4   00  ζ 0 SS11 11 U  U =()→⊕()1 :kerUUU⊥⊥ ranran,  0  ∗∗⊥∗⊥ V = []V1 0 :ranVV⊕( ran ) →( ker. V )

From the commutativity, we obtain

1 2 1 2 AAABBB21 =⊕21 21; 21 =⊕21 21. By simple computation, we get

∗∗ ∗∗1 1 2 2 2 3 1 2 4 2 U SV11 = USV111121,0 ASB11 21 === ASB21 11 21 ASB21 11 21 .

i i From the injectivity of A21 and B21: i = 1, 2, we derive that 2 3 4 S11 ===SS11 11 0.

∗ The injectivity of ABABA21,,,, 22 22 21 11, and B11 in the equalities 10 and 11 implies that

S12===S 21 S 220, S 13 R 2 = TS 2 31 = TS 2 33 R 2 = 0.

Setting EX11()= 11 AXB 11 11 11 − AXB 21 11 21. ∗∗ (i) If kerAA12∩={}=∩ ker 0 ker B12 ker B, then S13===SS 33 31 0 and therefore any operator SE∈ ker has the form

S11 00 S =  00S ,  23  00S32 

where S23 and S32 are arbitrary with

1 ∗∗1 ∗∗ ∥SSUSVUSV11 ∥ =∥11 ∥ =∥1111 ∥ =∥11 ∥ and

EX11()+ 11 S 11 ∗ ∗   ∥()+∥EX S = ∗∗S .  23  ∗∗S32 

Let S23=∣US 23 23∣, S32=∣SS 32 32∣ be the polar decomposition of S23 and S32, respectively, and set the operator ∗  0 U32 VI=⊕ ∗ .Then U23 0  EX11()+ 11 S 11 ∗ ∗   ∥EX ( )+ S ∥ ≥∥∥( VEX ( )+ S )∥ ≥ ∗∣∣∗S .  32   ∗∗∣∣S23 

Applying Lemma 5, we get

– =()B :

∥EX ( ) + S ∥ ≥max {∥ EX (11 ) + S 11 ∥ , ∥ S 32 ∥ , ∥ S 23 ∥} ≥ max {∥ S 11 ∥ , ∥ S 32 ∥ , ∥ S 23 ∥} = ∥ S ∥.

– =(Cp ): (1 ≤<∞p )

p p p 1//p p p p 1 p p ∥EX ( )+ S ∥p ≥(∥( EX11 )+ S 11 ∥p +∥ S 32 ∥p +∥ S 23 ∥p ) ≥(∥ S11 ∥p +∥ S 32 ∥p +∥ S 23 ∥p ) =∥ S ∥p . 44  Ahmed Bachir et al.

∗∗ (ii) If kerAA12∩≠{ ker 0} or kerBB12∩≠{ ker 0}, then any operator SE∈ ker has the form

SS110 13 S =  00S ,  23 SSS13 32 33

where S23 and S32 are arbitrary. By simple calculation, we have

 EX11()+ 11 S 11 ∗ AXR 11 13 2 + S 13   ∥()+∥EX S = ∗∗S .  23  TX23111 B++ S 3132 S TX 2332 R S 33

It is well known that the kernel of a quasinormal operator is a reduced subspace, then by application of Corollary 8, we obtain

∥+∥≥∥∥AXR11 13 2 S 13 kS 13

∥+∥≥∥∥TX23111 B S 31 kS 31

∥+∥≥∥∥TX2 33 R 2 S 33 kS 33 .

Therefore, by Lemma 6, we get

– =(B );

2 1  2 1 2 2 2 2 2  1 2 ∥()+∥≥EXSEXS ∥11 ( 11 )+ 11 ∥+ [∥∥+∥ SSSSS32 23 ∥+∥13 ∥+∥31 ∥+∥33 ∥]≥ ∥∥ S. 32  22  62

– =(Cp ): (2 ≤<∞p )

p 1 p 1 p p p p p ∥()+∥≥EXSS ∥11 ∥+ [∥∥+∥∥+∥∥+∥∥+∥∥] SSSSS32 23 13 31 33 p 3 p−−2 p 2p 2 p p p p p 1 1 1 ≥∥∥=∥∥SSp p . 2pp−−223 p 6 p − 2p

– =(Cp ): (12≤≤p ):

p 1 p 1 p p p p p ∥()+∥≥EXSS ∥11 ∥+ [∥∥+∥∥+∥∥+∥∥+∥∥] SSSSS32 23 13 31 33 p 32−−p p 22 p p p p p p 1 1 1 ≥∥∥=∥∥SSp p . 222−−pp3 p 6 p − 2p □

4 Conclusion

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, Theorem 2 was extended to non-normal operators including quasinormal, subnormal, and k-quasihyponormal operators. The main results are Theorems 12 and 18, both of considerable value in the relevant area of research, also the paper includes new ideas, along with a few new tools and techniques, and likely to attract considerable attention from researchers in operator theory and Banach space theory.

Acknowledgements: The authors would like to express their cordial gratitude to the referee for valuable comments which improved the paper. The authors would also like to add their great appreciation to the Deanship of Scientiﬁc Research at King Khalid University for funding this work through General Project Research, Grant number (GPR/247/42). Range-kernel weak orthogonality  45

Research funding: This research was funded through General Project Research, Grant number (GPR/ 247/42).

Author contributions: A. Bachir, A. Segres, and Nawal A. Sayyaf contributed to the design and implementa- tion of the research, to the analysis of the results, and to the writing of the manuscript.

Conﬂict of interest: Authors state no conﬂict of interest.

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