论文标题
一些硬质核矩阵的光谱
The spectrum of some Hardy kernel matrices
论文作者
论文摘要
对于$α> 0 $，我们考虑操作员$k_α\ colon \ ell^2 \ to \ ell^2 $对应于矩阵\ [\ left（\ frac {（nm）^{ - \ frac {1} {2}+α}}} {[\ max（n，m）]^{2α}} \ right）_ {n，m = 1}^\ infty。连续频谱与$ [0，2/α] $（多重性一）一致，并且没有奇异的连续频谱。连续频谱上方有有限数量的特征值。我们应用结果来证明，在Dirichlet系列$ \ Mathscr {H}^2 $的耐力空间上，复制的内核论文不满意。
For $α> 0$ we consider the operator $K_α\colon \ell^2 \to \ell^2$ corresponding to the matrix \[\left(\frac{(nm)^{-\frac{1}{2}+α}}{[\max(n,m)]^{2α}}\right)_{n,m=1}^\infty.\] By interpreting $K_α$ as the inverse of an unbounded Jacobi matrix, we show that the absolutely continuous spectrum coincides with $[0, 2/α]$ (multiplicity one), and that there is no singular continuous spectrum. There is a finite number of eigenvalues above the continuous spectrum. We apply our results to demonstrate that the reproducing kernel thesis does not hold for composition operators on the Hardy space of Dirichlet series $\mathscr{H}^2$.
