Elementary operators and orthogonality. (English) Zbl 1084.47510

J. Anderson [Proc.Am.Math.Soc.38, 135–140 (1973; Zbl 0255.47036)] defined a subspace \(M\) of a normed space \(E\) to be orthogonal to a subspace \(N\subseteq E\) if \(\|m+n\|\geq\|n\|\) for all \(m\in M\), \(n\in N\). He proved that the range of an inner derivation \(\delta_A\) on \(B(l^2)\), where \(A\) is a normal operator on \(l^2\), is orthogonal to the kernel of \(\delta_A\).
Let \(A_1,\dots,A_n\), \(B_1,\dots,B_n\) be (bounded, linear) operators on \(l^2\) and define the associated elementary operator \(S\) on \(B(l^2)\) by \(X\mapsto\sum_{i=1}^n A_iXB_i\). The author discusses criteria under which the range of \(S-\text{id}\) is orthogonal to the kernel of \(S-\text{id}\), when considered as an operator on \(B(l^2)\) or on the von Neumann–Schatten classes.


47B47 Commutators, derivations, elementary operators, etc.
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)


Zbl 0255.47036
Full Text: DOI


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