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Quasi-normed operator ideals on Banach spaces and interpolation. (English) Zbl 1052.46061
The authors apply the \(K\)-method of real interpolation [see, e.g., C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press (1988; Zbl 0647.46057)] to the quasi-normed operator ideals on Banach spaces [see, e.g., A. Pietsch, Operator Ideals, Berlin (1978; Zbl 0399.47039)] to show how new operator ideals can be obtained. In particular, they prove the following variant of reiteration theorem: Let \((\mathcal A,a)\) and \((\mathcal B,b)\) be two quasi-normed operator ideals on Banach spaces, and \(\mathcal C_{\theta_0,p_0}=(\mathcal A,\mathcal B)_{\theta_0,p_0}\), \(\mathcal C_{\theta_1,p_1}=(\mathcal A,\mathcal B)_{\theta_1,p_1}\), where \(0<\theta_i<1\), \(1\leq p_i<\infty\), \(i=0,1\). Then \((\mathcal C_{\theta_0,p_0},\mathcal C_{\theta_1,p_1})_{\eta,p}=\mathcal C_{\eta,p}\), with equivalent norms, where \(\theta=(1-\eta)\theta_0+\eta\theta_1\), \(0<\eta<1\), \(1\leq p<\infty\).
46M35 Abstract interpolation of topological vector spaces
47L20 Operator ideals
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