Research Article Range-Kernel Orthogonality and Elementary Operators: The Nonsmoothness Case A. Bachir , 1 A. Segres, 2 and Nawal Sayyaf 3 1 Department of Mathematics, University of King Khalid, Abha, Saudi Arabia 2 University of Mascara, Mascara, Algeria 3 Department of Mathematics, College of Science, University of Bisha, Bisha, Saudi Arabia Correspondence should be addressed to A. Bachir; abishr@kku.edu.sa Received 3 December 2019; Accepted 25 May 2020; Published 24 June 2020 Academic Editor: Eulalia Mart´ ınez Copyright © 2020 A. Bachir et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e characterization of the points in C 1 (H), the trace class operators, that are orthogonal to the range of elementary operators has been carried out for certain kinds of elementary operators by many authors in the smooth case. In this note, we study that the characterization is a problem in nonsmoothness case for general elementary operators, and we give a counter example to S. Mecheri and M. Bounkhel results. 1. Introduction Let B(H) be the algebra of all bounded linear operators acting on a complex separable Hilbert space H and A ∈ B(H) be compact operator, and let s 1 (A) ≥ s 2 (A) ≥ ··· ≥ 0 denote the eigenvalues of |A|�(A ∗ A) 1/2 arranged in their decreasing order. A ∈ C 1 (H), the class trace, if ‖A‖ 1 �  ∞ i�1 s i (A)� tr|A| <∞, (1) where tr denotes the trace functional [1]. We recall the definition of Birkhoff–James’s orthogo- nality in Banach spaces [2, 3]. Definition 1. If A is a complex Banach space, then for any elements a, b ∈ A, we say that a is orthogonal to b, noted by a⊥b, iff for all λ, β ∈ C there holds ‖λb + βa‖ ≥ ‖βa‖. (2) If M and N are linear subspaces in A, we say that M is orthogonal to N, noted by M⊥N, if ‖a + b‖ ≥ ‖a‖ for all a ∈ M and all b ∈ N. If M � span a {}, we simply write a⊥N. (i) e orthogonality in this sense is not symmetric (ii) If A isaHilbertspacewithitsinnerproduct 〈·〉,then it follows from (2) that 〈a, b〉� 0 which means that Birkhoff–James’s orthogonality generalizes the usual sense in Hilbert space We also recall the definition of the range-kernel or- thogonality for a pair of operators (E, T) on Banach spaces introduced by R. Harte [4]. Definition 2. If E: A ⟶ Y and T: B ⟶ C are bounded linear operators between Banach spaces. T is called range- kernel orthogonal to E provided kerT is orthogonal to ranE in the sense of the Definition 1, i.e., s ∈ ker T ⟹ ‖s + E(x)‖ ≥ ‖s‖, forall x ∈ A. (3) If T � E, we shall call that E is orthogonal. e elementary operator is an operator E defined on Banach (A, B)-bimodule M with its representation E(x)�  n i�1 a i xb i , where a �(a i ) i ∈ A n and b �(b i ) i ∈ B n are n-tuples of algebra elements. e length of E is defined to be the smallest number of multiplication terms required for any representation  j a j xb j for E. e case of the elementary operator is restricted to the operator δ A,B ∈ B[B(H)], which is well known, as generalized derivation induced by fixed Hindawi Complexity Volume 2020, Article ID 5657678, 7 pages https://doi.org/10.1155/2020/5657678