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Structural properties of elementary operators. (English) Zbl 0627.47015

Let \({\mathcal A}\) and \({\mathcal B}\) denote complex Banach algebras and let \({\mathcal M}\) be a left Banach \({\mathcal A}\)-module and a right Banach \({\mathcal B}\)-module. If \(A=(A_ 1,...,A_ n)\in {\mathcal A}^{(n)}\) and \(B=(B_ 1,...,B_ n)\in {\mathcal B}^{(n)}\) the elementary operator R(A,B) acting on \({\mathcal M}\) is defined by \(R(A,B)(X)=\sum^{n}_{i=1}A_ iXB_ i\). This paper studies these operators in the following settings:
1) \({\mathcal H}_ 1\) and \({\mathcal H}_ 2\) are separable infinite dimensional Hilbert spaces, \({\mathcal A}={\mathcal L}({\mathcal H}_ 2)\), \({\mathcal B}={\mathcal L}({\mathcal H}_ 1)\) and \({\mathcal M}={\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 2);\)
2) \({\mathcal M}=\tilde {\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 2)\equiv {\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 2)/{\mathcal K}({\mathcal H}_ 1,{\mathcal H}_ 2)\) where \({\mathcal K}({\mathcal H}_ 1,{\mathcal H}_ 2)\) is the space of compact operators from \({\mathcal H}_ 1\) to \({\mathcal H}_ 2\), \({\mathcal A}=\tilde {\mathcal L}({\mathcal H}_ 1)\equiv {\mathcal L}({\mathcal H}_ 1)/{\mathcal K}({\mathcal H}_ 1)\) and \({\mathcal B}=\tilde {\mathcal L}({\mathcal H}_ 2)\equiv {\mathcal L}({\mathcal H}_ 2)/{\mathcal K}({\mathcal H}_ 2).\)
The authors describe the elementary operators \(R(\tilde A,\tilde B)\) on \(\tilde {\mathcal L}({\mathcal H}_ 1,{\mathcal H}_ 2)\) which are pseudodiagonal, pseudoalgebraic or which have the strong spectral splitting property (s.s.s.p.). In general, for a Banach space \({\mathcal X}\), \(T\in {\mathcal L}({\mathcal X})\) has the s.s.s.p. if \(\cup_{\lambda \in C}{\mathcal X}_{\lambda}(T)\) generates a linear manifold which is dense in \({\mathcal X}\), where \({\mathcal X}_{\lambda}(T)=\{x\in {\mathcal X}:\lim_{k\to \infty}\| (T-\lambda)^ kx\|^{1/k}=0\}.\)
Another result of the paper is the proof of the following conjecture of C. K. Fong and A. R. Sourour [Can. J. Math. 31, 845-857 (1979; Zbl 0368.47024)]: there is no nonzero compact elementary operator on the Calkin algebra \({\mathcal L}({\mathcal H})/{\mathcal K}({\mathcal H})\).
Reviewer: G.Corach

MSC:

47B47 Commutators, derivations, elementary operators, etc.
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
47L10 Algebras of operators on Banach spaces and other topological linear spaces

Citations:

Zbl 0368.47024
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