r-Summing Hankel Operators in Function Spaces

• The paper establishes that an r-summing Hankel operator on Fock spaces is equivalent to having its symbol in the IDA space, linking the r-summing norm with localized L^p oscillations.
• Key methodologies include Rademacher function techniques, Khintchine inequalities, and Carleson embedding estimates to derive precise norm equivalences.
• Applications extend to weighted Bergman spaces, illustrating the Berger–Coburn phenomenon and symmetry properties, and highlighting broader operator ideal implications.

An $r$-summing Hankel operator is a bounded linear operator of Hankel type, acting between function spaces, that satisfies a quantitative summability condition indexed by $r \in [1, \infty)$. These operators generalize the classical notions of compactness and Schatten class in operator theory, and their $r$-summing norm is closely related to localized $L^p$ oscillations of the symbol function and to intrinsic function space structures. This article presents the primary definitions, structural characterizations, and main theorems for $r$-summing Hankel operators on both Fock and Bergman spaces, including a discussion of associated phenomena and applications.

1. Fock Spaces and Hankel Operators

Let $\alpha > 0$ and $1 \leq p < \infty$. The Gaussian weighted Lebesgue space on $\mathbb{C}^n$ is

$$L^p_\alpha = \left\{f \ \text{measurable} : \|f\|_{p,\alpha}^p = \int_{\mathbb{C}^n} |f(z)|^p e^{-\frac{\alpha p}{2}|z|^2} dv(z) < \infty \right\},$$

where $dv$ denotes Lebesgue measure. The holomorphic Fock space is then $$F^p_\alpha = \{f \in L^p_\alpha : f \text{ is entire}\}$$, a Banach space for $p \geq 1$.

Given a symbol $f$ such that $fg \in L^p_\alpha$ for all $g \in F^p_\alpha$, the Hankel operator on Fock space is defined by

$$H_f : F^p_\alpha \to L^p_\alpha, \quad H_f(g) = f g - P(fg),$$

where $P$ is the orthogonal projection from $L^2_\alpha$ onto $F^2_\alpha$, with kernel $K(z, w) = e^{\alpha \langle z, w \rangle}$ and normalized kernel $k_z$.

For weighted Bergman spaces on the unit ball $\mathbb{B}_n \subset \mathbb{C}^n$, define

$$A^p_\alpha = \left\{ f \text{ holomorphic on } \mathbb{B}_n : \|f\|_{p,\alpha}^p = \int_{\mathbb{B}_n} |f(z)|^p dv_\alpha(z) < \infty \right\}, \quad dv_\alpha(z) = c_\alpha (1 - |z|^2)^\alpha dv(z).$$

Given $f \in L^1(dv_\beta)$, the big Hankel operator $H_f^\beta$ and the little Hankel operator $h_f^\beta$ are defined via projections $P_\beta$, $\overline{P}_\beta$ acting on $L^q(dv_\beta)$.

2. Definition and Norm of $r$-Summing Operators

A bounded operator $T: X \to Y$ between Banach spaces is called absolutely $r$-summing ($T \in \Pi_r(X, Y)$) if there exists $C < \infty$ such that for every finite sequence $\{x_k\} \subset X$,

$$\left( \sum_{k=1}^N \|T x_k\|_Y^r \right)^{1/r} \leq C \sup_{\phi \in B_{X^*}} \left( \sum_{k=1}^N |\phi(x_k)|^r \right)^{1/r}.$$

The infimum of all such $C$ is the $r$-summing norm $\pi_r(T)$. By Pietsch factorization, the $r$-summing property admits a probabilistic domination:

$$\|T x\|_Y \leq \pi_r(T) \left( \int_{B_{X^*}} |\phi(x)|^r d\mu(\phi) \right)^{1/r}$$

for some probability measure $\mu$ on the dual unit ball.

3. IDA Spaces, Oscillation Norms, and Characterization

For $f \in L^p_{\text{loc}}(\mathbb{C}^n)$ and $r > 0$, define the local approximation error

$$G_{p,r}(f)(z) := \inf_{h \in \text{Hol}(B(z,r))} \left( \frac{1}{|B(z,r)|} \int_{B(z,r)} |f(w) - h(w)|^p dv(w) \right)^{1/p},$$

where $B(z, r)$ is the ball of radius $r$. For $0 < s \leq \infty$, set

$$\text{IDA}^{s,p}_r = \left\{ f : \|f\|_{\text{IDA}^{s,p}_r} := \|G_{p,r}(f)\|_{L^s(\mathbb{C}^n)} < \infty \right\}.$$

Norms for different $r > 0$ are equivalent; typically $r = 1$ is used. The crucial exponent $\kappa = \kappa(p, r)$ is defined by the piecewise formula: $$\kappa(p, r) = \begin{cases} 2, & 1 \leq p \leq 2, r \geq 1, \ p' = \frac{p}{p - 1}, & p \geq 2, 1 \leq r \leq p', \ r, & p \geq 2, p' \leq r \leq p, \ p, & p \geq 2, r \geq p. \ \end{cases}$$

4. Main Theorem: Equivalence of $r$-Summing Norm and IDA-Norm

For $1 \leq p, r < \infty$ and $\kappa = \kappa(p, r)$, the following equivalence holds:

$$H_f \in \Pi_r(F^p_\alpha, L^p_\alpha) \iff f \in \text{IDA}^{\kappa, p}$$

with two-sided estimates

$$C_1 \|f\|_{\text{IDA}^{\kappa, p}} \leq \pi_r(H_f : F^p_\alpha \to L^p_\alpha) \leq C_2 \|f\|_{\text{IDA}^{\kappa, p}}$$

for constants $C_1, C_2$ depending on $p, r, \alpha, n$ (Hu et al., 3 Jan 2026). The proof bifurcates into three regimes according to $(p, r)$ and exploits Rademacher function techniques, Khintchine inequalities, and cotype-based decompositions. The reverse direction utilizes a decomposition into holomorphic plus error parts with estimates based on Carleson embedding characterizations.

For weighted Bergman spaces, analogous theorems hold. The $r$-summing norm of the big Hankel operator $H_f^\beta$ is given by

$$\pi_r(H_f^\beta) \simeq \left\| (1 - |z|^2)^\gamma G_{q, \delta}(f)(z) \right\|_{L^s(\mathbb{B}_n, d\lambda) }$$

where $$\gamma = \frac{n+1+\beta}{q} - \frac{n+1+\alpha}{p}$$, and $s$ depends on $(p, q, r)$ in a piecewise manner (Fan et al., 27 Nov 2025).

5. Berger-Coburn Phenomenon and Symmetry Properties

A phenomenon particular to $r$-summing Hankel operators is the Berger–Coburn phenomenon (BCP): For the operator ideal $\mathcal{M}$ on $L^\infty$ symbols, BCP holds if for every bounded symbol $f$,

$$H_f \in \mathcal{M} \iff H_{\bar{f}} \in \mathcal{M}.$$

The key estimate (Proposition 4.1 in (Hu et al., 3 Jan 2026)) gives

$$f \in L^\infty \cap \text{IDA}^{\kappa, p} \Longrightarrow \bar{f} \in \text{IDA}^{\kappa, p}, \quad \|\bar{f}\|_{\text{IDA}^{\kappa, p}} \lesssim \|f\|_{\text{IDA}^{\kappa, p}},$$

yielding

$$H_f \in \Pi_r \iff f \in \text{IDA}^{\kappa, p} \iff \bar{f} \in \text{IDA}^{\kappa, p} \iff H_{\bar{f}} \in \Pi_r$$

with norm equivalence $\pi_r(H_f) \simeq \pi_r(H_{\bar{f}})$. Thus BCP holds for all $r$-summing Hankel operators.

6. Special Cases and Corollaries

• For $p = r = 2$ (Hilbert–Schmidt case), $\kappa = 2$. Then $H_f \in \Pi_2(F^2_\alpha \to L^2_\alpha)$ iff $f \in \text{IDA}^{2, 2}$, and $\pi_2(H_f) \simeq \| G_2(f) \|_{L^2}$.
• For $p = 2$, any $r \geq 1$: $\Pi_r(F^2_\alpha \to L^2_\alpha) = \Pi_2$; again, $\kappa = 2$.
• For $p = 1$, $r$ arbitrary: Retrieve characterization of $1$-summing Hankel operators in terms of the $L^2$ IDA-norm.
• For Bergman spaces, when $p = q$, $\gamma = (\beta - \alpha)/p$ and $\kappa=2/p$; e.g., for $1 < p < 2$, $r = 1$, $H_f^\beta$ is absolutely summing iff $$(1 - |z|^2)^{(\beta - \alpha)/p} G_{p, \delta}(f) \in L^{2}(\mathbb{B}_n, d\lambda)$$.

7. Extensions and Further Remarks

All results for Fock spaces extend verbatim to general Fock-type spaces $F^p_\varphi$ under the uniform convexity condition $\text{Hess}_\mathbb{R} \varphi \simeq I$ [HV22, HV23]. Analogous characterizations are expected for doubling Fock spaces (Christ–Massaneda–Ortega-Cerdà) and for Toeplitz operators (see Hu–Wang, arXiv (Hu et al., 24 Sep 2025)).

The scale of IDA spaces interpolates between classical function spaces such as BMO and the limiting compactness/Schatten class characterizations as $r \to \infty$, $p \to \infty$. The methods—Pietsch’s domination theorem and Khintchine inequalities—are robust and may be applicable to other reproducing kernel Hilbert/Banach settings.

A plausible implication is that $r$-summing characterizations provide a unified framework for absolute summability, covering both trace and compactness properties, and distinguishing operator ideals in holomorphic function space settings (Hu et al., 3 Jan 2026, Fan et al., 27 Nov 2025).