Extremal Non-Compactness of Composition Operators with Linear Fractional Symbol

We realize norms of most composition operators acting on the Hardy space with linear fractional symbol as roots of hypergeometric functions. This realization leads to simple necessary and sufficient conditions on the symbol to exhibit extremal non-compactness, establishes equivalence of cohyponormality and cosubnormality of composition operators with linear fractional symbol, and yields a complete classification of those linear fractional that induce composition operators whose norms are determined by the action of the adjoint on the normalized reproducing kernels in the Hardy space

The spectrum of some Hardy kernel matrices

For $\alpha > 0$ we consider the operator $K_\alpha \colon \ell^2 \to \ell^2$ corresponding to the matrix $$\left(\frac{(nm)^{-\frac{1}{2}+\alpha}}{[\max(n,m)]^{2\alpha}}\right)_{n,m=1}^\infty.$$ By interpreting $K_\alpha$ as the inverse of an unbounded Jacobi matrix, we show that the absolutely continuous spectrum coincides with $[0, 2/\alpha]$ (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 $\mathscr{H}^2$.Comment: Minor changes. Theorem C (iii) is ne

Norms of Composition Operators on the Hardy Space

this paper. For ff 2 U , the reproducing kernel at ff, denoted k ff , is defined b