Exact constants in inequalities for norms of powers of linear operators in a finite-dimensional space (Q1326025)

Let \(A\) be a square matrix of order \(m\times m\), \(0\leq k\leq n\), \(1\leq p,q,r\leq\infty\), \(0\leq\alpha\leq 1\). The Landau ratio is defined by the relation \[ L(x,A,n,k,p,q,r,\alpha)=\| A^{n-k}x\|_ q/ (\| A^ n x\|^ \alpha_ p\| x\|_ r^{1-\alpha}). \] The following problems are considered: To find the estimation of the smallest \(\hat C\) and the greatest \(\check C\) for which the inequalities \[ \| A^{n-k}x\|_ q\leq\hat C\| A^ n x\|_ p^ \alpha\| x\|_ r^{1-\alpha} \quad \text{and} \quad \| A^ {n-k} x\|_ q \geq \check C \| A^ n x\|^ \alpha_ p \| x\|_ r^{1-\alpha} \] hold for all \(x\).