The Apostol-Fialkow formula for elementary operators on Banach spaces
Let \(E\) be a real or complex Banach space. Any two \(n\)-typles \(A= (A_1,\dots, A_n)\) and \(B= (B_1,\dots, B_n)\) of bounded linear operators on \(E\) determine an elementary operator \({\mathbf E}_{A,B}: L(E)\to L(E)\) by \({\mathbf E}_{A,B}(S)= \sum_{1\leq i\leq n} A_i SB_i\), \(\forall S\in L(E)\). And the induced elementary operator \({\mathbf e}_{a,b}\) on the Calkin algebra \({\mathbf C}(E)\equiv L(E)/K(E)\) is defined by \({\mathbf e}_{a,b}(s)= \sum_{1\leq i\leq n} a_isb_i\), \(\forall s\in{\mathbf C}(E)\), \(s= S+ K(E)\), \(S\in L(E)\), where \(K(E)\) is the space of compact operators on \(E\). Let \(E\) be a Banach space with a Schauder basis \((e_k)_{k\geq 1}\), the natural basis projections \((P_k)_{k\geq 1}\) and \((Q_k)_{k\geq 1}\) are defined by \(P_k(\sum_{i\geq 1} a_ie_i)= \sum_{1\leq i\leq k} a_ie_i\), \(Q_k= I- P_k\), \(k\geq 1\). The basis \((e_k)_{k\geq 1}\) is called unconditional if \(\sum_{i\geq 1}\lambda_i a_ie_i\) converges in \(E\) whenever \(\sum_{i\geq 1} a_ie_i\in E\) and \((\lambda_i)\in \ell^\infty\); the unconditional basis constant of \((e_k)\) is \(C= \sup\{\|M_\theta\|; \theta= (\theta_i)\in \{-1,1\}^{\mathbb{N}}\}\), where \(M_\theta (\sum_{i\geq 1} a_ie_i)= \sum_{i\geq 1} \theta_ia_ie_i\), \(\sum_{i\geq 1} a_ie_i\in E\). In this paper, the authors prove the following main result, which improves some results of Apostol and Fialkow, and Magajna. Let \(E\) be a Banach space with an unconditional basis \((e_k)\) and an unconditional constant \(C\). Then we have: \[ \begin{aligned} \|{\mathbf e}_{a,b}:{\mathbf C}(E)\to {\mathbf C}(E)\|_w &\geq C^{-4}\|{\mathbf e}_{a,b}:{\mathbf C}(E)\to {\mathbf C}(E)\|,\\ \|{\mathbf E}_{A,B}: L(E)\to L(E)\|_w &\geq C^{-4}\|{\mathbf e}_{a,b}:{\mathbf C}(E)\to {\mathbf C}(E)\|.\end{aligned} \] If \(E= \ell^p\), \(1< p< \infty\), then: \(\|{\mathbf E}_{A,B}\|_w=\|{\mathbf e}_{a,b}\|= \|{\mathbf e}_{a,b}\|_w\).
scientific article; zbMATH DE number 1271580
scientific article