Title: Applications of the theory of s.n. functions to the duality of analytic function spaces and the Hankel operators in $\mathfrak{S}$

Authors: Mark C. Ho and Mu Ming Wong

Issue: Volume 55 (2006), Issue 5, 1645-1670

Abstract: Let $\mathcal{H}$ be a separable Hilbert space, and $T$ be a bounded operator on $\mathcal{H}$. The singular values of $T$ are the numbers defined by $s_{n}(T) = \inf\{\|T - S\|: \mbox{rank}(S) < n\}$, $n = 1,2,\dots$. Let $\Pi = \{\pi_{n}\}$ be a nonincreasing sequence of nonnegative numbers. Then a compact operator $T$ is said to be in the ideals $\mathfrak{S}_{\Pi}^{(0)}$, $\mathfrak{S}_{\Pi}$ and $\mathfrak{S}_{\pi}$ if $\sum_{1}^{N}s_{n}(T) = o(\sum_{1}^{N}\pi_n)$, $\sum_{1}^{\infty}s_{n}(T) = O(\sum_{1}^{N}\pi_n)$, and $\sum_{1}^{\infty}\pi_{n}s_{n}(T) < \infty$, respectively. Ideals such as $\mathfrak{S}_{\Pi}^{(0)}$, $\mathfrak{S}_{\Pi}$ and $\mathfrak{S}_{\pi}$ are examples in a more general class of ideals of compact operators on $\mathcal{H}$ called the symmetrically normed ideals (or s.n. ideals, in brief), which are generated by the so-called symmetric norming functions (or s.n. functions). In this article we focus on operators on concrete spaces (in particular, the Hankel operators) belonging to these s.n. ideals and, in particular, in $\mathfrak{S}_{\Pi}^{(0)}$, $\mathfrak{S}_{\Pi}$ and $\mathfrak{S}_{\pi}$, under certain conditions of $\Pi$. Moreover, we will use the s.n. functions, together with function theory to construct concrete function spaces that correspond to these ideals.