Abstract
The characterization of the points in C1ℋ, 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.
Highlights
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 s1(A) ≥ s2(A) ≥ · · · ≥ 0 denote the eigenvalues of |A| (A∗A)1/2 arranged in their decreasing order
We recall the definition of Birkhoff–James’s orthogonality in Banach spaces [2, 3]
If A is a complex Banach space, 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
Summary
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 s1(A) ≥ s2(A) ≥ · · · ≥ 0 denote the eigenvalues of |A| (A∗A)1/2 arranged in their decreasing order. We recall the definition of Birkhoff–James’s orthogonality in Banach spaces [2, 3]. If A is a complex Banach space, 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. We recall the definition of the range-kernel orthogonality for a pair of operators (E, T) on Banach spaces introduced by R. If E: A ⟶ Y and T: B ⟶ C are bounded linear operators between Banach spaces. Where ker δA,B denotes the kernel of δA,B. is means that the kernel of δA,B is orthogonal to its range. E main purpose of this paper is to characterize the elements that are orthogonal to the range of arbitrary elementary operators defined on C1(H) in nonsmoothness case and give a counter example to the results of S. In addition to the notations and the definitions already introduced, we set, if X is a normed linear space over a field K R or C, we denote by B(X) the space of all linear bounded operator on X, the closure of the range of an operator T ∈ B(X) will be denoted by ran(T), the restriction of T to an invariant subspace M will be denoted by T|M, and the commutator AB − BA of the operators A, B will be denoted by [A, B]
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