Range-kernel weak orthogonality of some elementary operators

where tr denotes the trace functional. In case p = ∞, we denote by ( ) ∞ , the ideal of compact operators equipped with the norm X s X 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 operatorT B ∈ ( ) will be denoted by T ran and T ker denotes the kernel ofT . The restriction ofT to an invariant subspace will be denoted byT ∣ , and the commutator AB BA − of the operators A B , will be denoted by A B , [ ]. We recall the definition of Birkhoff-James’s orthogonality in Banach spaces [2,3].


Introduction
Let B ( ) be the algebra of all bounded linear operators acting on a complex separable Hilbert space . Given A B B , ∈ ( ), we define the generalized derivation δ B B : where tr denotes the trace functional. In case p = ∞, we denote by ( ) ∞ , the ideal of compact operators equipped with the norm X s X 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 operator T B ∈ ( ) will be denoted by T ran and T ker denotes the kernel of T . The restriction of T to an invariant subspace will be denoted by T | , and the commutator AB BA − of the operators A B , will be denoted by A B , [ ]. We recall the definition of Birkhoff-James's orthogonality in Banach spaces [2,3].
If is a complex Banach space, then for any elements x y , ∈ , we say that x is orthogonal to y, noted by x y ⊥ , iff for all α β , ∈ there holds αy βx βx . ∥ + ∥ ≥∥ ∥ (2) for all α β , ∈ or .
are n-tuples of algebra elements. The length of E is defined to be the smallest number of multiplication terms required for any representation a xb j j j ∑ for E.
In this note, we consider B = = ( ) and B = ( ) or p : 1 p = ( ≤ <∞) and the length of E will be less or equal to 2, i.e., if A A A B B B , , , This means that the kernel of δ A 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 δ A B , | is orthogonal to δ ker A B , ∩ . A detailed study of range-kernel orthogonality for generalized derivation δ A B , has received much attention in recent years and has been carried out in a large number of studies [3,5,[7][8][9][10][11][12][13] and are based on the following result.
D. Keckic [14] and A. Turnšek [15] extended Theorem 2 to the elementary operator E defined by E X AXB ) and B D , ( ) are 2-tuples of commuting normal operators. Duggal [16] generalized the famous theorem to the case A C , ( )and B D , ( )are 2-tuples of commuting operators, where A B , 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 k We say that T is weakly orthogonal to E, written T E ∠ , or equivalently , consequently we get a k 1 -gap between the subspaces E ker and E ran , which corresponds to the "range-kernel weak-orthogonality" for an operator E. If k 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 E ker and E ran .
is a positive operator. Furthermore, we have the following proper inclusion The n-tuple A is said to be normal if A is commuting and each A i n 1, , i ( = … ) 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.
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: Then Using Lemma 5(2), we get a simple form of the previous result as follows.
In the sequel ξ denotes the elementary operator defined by Proof. The proof is the same as the one in Theorem 4. □

Proposition 10. Let A B
, be doubly commuting operators in B ( ) and Proof. We consider the following three cases: , then with respect to the decomposition and from the hypothesis it yields where A 2 is an injective operator. Hence, and the other entries are arbitrary. Choosing X and S as follows: 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 , we proceed similarly as in the first case, it suffices to replace A by B and B by A in the preceding argument.
, then with respect to the decomposition and the other entries are arbitrary. Choosing X and S as where e and R are as in (i). Then Since A 1 is injective and S ξ ker ∈ , we obtain To conclude the proof we can argue similarly as in the first case (i). □ , a n d 0 0 0 . Proof.

ξ S A S A A S A A S A
Let us now finish the proof for the elementary operator E X A XA N XN . Consider the space ⊕ and define the following operators on B ( ⊕ ) as , a n d 0 0 0 .   , 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.
, then E is w-orthogonal and E Ẽ ∠ . Furthermore, E and Ẽ satisfy assertions (i), (ii), and (iii) cited in Theorem 12, in each of the following cases: (i) A 1 and B 1 * are hyponormal operators; (ii) A 1 is k-quasihyponormal and B 1 * 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?
, then for all X B ∈ ( ), we have , then E is w-orthogonal and E Ẽ ∠ .
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. □  , and R 2 * are quasinormal operators and T C R C , , , ( ) ( ) are 2-tuples of doubly commuting operators.
Since T ker 1 reduces T 1 (resp. R ker 1 * reduces R 1 * ) and by commutativity, we have that