On the matrix norm subordinate to the Hölder norm (Q1301241)

For a matrix \(M\in {\mathbb C}^{m\times n}\) let \(\|M\|_p\) be the standard subordinate matrix norm generated by the vector \(\ell_p\) norm. Let \(P\in {\mathbb R}^{m\times n}_+\), \(p\in (1,\infty)\), \(1/p+1/q=1\). Let also a homogeneous strongly monotone operator \(T:{\mathbb R}^n_+\to {\mathbb R}^n_+\) be defined as \((Tv)_j=[\sum^m_{i=1}p_{ij}(Pv)_i^{p-1}]^{q-1}\), \(j=1,\dots,n\). The authors obtain interrelations between the norm \(\|P\|_p\) and the following eigenvalue problem \(T\alpha=\lambda\alpha\). Namely, assume that the eigenvalue problem \(T\alpha=\lambda\alpha\) has an eigenvector \(\alpha\) with positive components only, corresponding to a positive eigenvalue~\(\lambda\). Then \(\|P\|_p=\lambda ^{1/q}\).