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 $r \in [1, \infty)$0, a Banach space for $r \in [1, \infty)$1.

Given a symbol $r \in [1, \infty)$2 such that $r \in [1, \infty)$3 for all $r \in [1, \infty)$4, the Hankel operator on Fock space is defined by

$r \in [1, \infty)$5

where $r \in [1, \infty)$6 is the orthogonal projection from $r \in [1, \infty)$7 onto $r \in [1, \infty)$8, with kernel $r \in [1, \infty)$9 and normalized kernel $r$0.

For weighted Bergman spaces on the unit ball $r$1, define

$r$2

Given $r$3, the big Hankel operator $r$4 and the little Hankel operator $r$5 are defined via projections $r$6, $r$7 acting on $r$8.

2. Definition and Norm of $r$9-Summing Operators

A bounded operator $L^p$0 between Banach spaces is called absolutely $L^p$1-summing ($L^p$2) if there exists $L^p$3 such that for every finite sequence $L^p$4,

$L^p$5

The infimum of all such $L^p$6 is the $L^p$7-summing norm $L^p$8. By Pietsch factorization, the $L^p$9-summing property admits a probabilistic domination:

$r$0

for some probability measure $r$1 on the dual unit ball.

3. IDA Spaces, Oscillation Norms, and Characterization

For $r$2 and $r$3, define the local approximation error

$r$4

where $r$5 is the ball of radius $r$6. For $r$7, set

$r$8

Norms for different $r$9 are equivalent; typically $\alpha > 0$0 is used. The crucial exponent $\alpha > 0$1 is defined by the piecewise formula: $\alpha > 0$2

4. Main Theorem: Equivalence of $\alpha > 0$3-Summing Norm and IDA-Norm

For $\alpha > 0$4 and $\alpha > 0$5, the following equivalence holds:

$\alpha > 0$6

with two-sided estimates

$\alpha > 0$7

for constants $\alpha > 0$8 depending on $\alpha > 0$9 (Hu et al., 3 Jan 2026). The proof bifurcates into three regimes according to $1 \leq p < \infty$0 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 $1 \leq p < \infty$1-summing norm of the big Hankel operator $1 \leq p < \infty$2 is given by

$1 \leq p < \infty$3

where $1 \leq p < \infty$4, and $1 \leq p < \infty$5 depends on $1 \leq p < \infty$6 in a piecewise manner (Fan et al., 27 Nov 2025).

5. Berger-Coburn Phenomenon and Symmetry Properties

A phenomenon particular to $1 \leq p < \infty$7-summing Hankel operators is the Berger–Coburn phenomenon (BCP): For the operator ideal $1 \leq p < \infty$8 on $1 \leq p < \infty$9 symbols, BCP holds if for every bounded symbol $\mathbb{C}^n$0,

$\mathbb{C}^n$1

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

$\mathbb{C}^n$2

yielding

$\mathbb{C}^n$3

with norm equivalence $\mathbb{C}^n$4. Thus BCP holds for all $\mathbb{C}^n$5-summing Hankel operators.

6. Special Cases and Corollaries

• For $\mathbb{C}^n$6 (Hilbert–Schmidt case), $\mathbb{C}^n$7. Then $\mathbb{C}^n$8 iff $\mathbb{C}^n$9, and $$L^p_\alpha = \left\{f \ \text{measurable} : \|f\|_{p,\alpha}^p = \int_{\mathbb{C}^n} |f(z)|^p e^{-\frac{\alpha p}{2}|z|^2} dv(z) < \infty \right\},$$0.
• For $$L^p_\alpha = \left\{f \ \text{measurable} : \|f\|_{p,\alpha}^p = \int_{\mathbb{C}^n} |f(z)|^p e^{-\frac{\alpha p}{2}|z|^2} dv(z) < \infty \right\},$$1, any $$L^p_\alpha = \left\{f \ \text{measurable} : \|f\|_{p,\alpha}^p = \int_{\mathbb{C}^n} |f(z)|^p e^{-\frac{\alpha p}{2}|z|^2} dv(z) < \infty \right\},$$2: $$L^p_\alpha = \left\{f \ \text{measurable} : \|f\|_{p,\alpha}^p = \int_{\mathbb{C}^n} |f(z)|^p e^{-\frac{\alpha p}{2}|z|^2} dv(z) < \infty \right\},$$3; again, $$L^p_\alpha = \left\{f \ \text{measurable} : \|f\|_{p,\alpha}^p = \int_{\mathbb{C}^n} |f(z)|^p e^{-\frac{\alpha p}{2}|z|^2} dv(z) < \infty \right\},$$4.
• For $$L^p_\alpha = \left\{f \ \text{measurable} : \|f\|_{p,\alpha}^p = \int_{\mathbb{C}^n} |f(z)|^p e^{-\frac{\alpha p}{2}|z|^2} dv(z) < \infty \right\},$$5, $$L^p_\alpha = \left\{f \ \text{measurable} : \|f\|_{p,\alpha}^p = \int_{\mathbb{C}^n} |f(z)|^p e^{-\frac{\alpha p}{2}|z|^2} dv(z) < \infty \right\},$$6 arbitrary: Retrieve characterization of $$L^p_\alpha = \left\{f \ \text{measurable} : \|f\|_{p,\alpha}^p = \int_{\mathbb{C}^n} |f(z)|^p e^{-\frac{\alpha p}{2}|z|^2} dv(z) < \infty \right\},$$7-summing Hankel operators in terms of the $$L^p_\alpha = \left\{f \ \text{measurable} : \|f\|_{p,\alpha}^p = \int_{\mathbb{C}^n} |f(z)|^p e^{-\frac{\alpha p}{2}|z|^2} dv(z) < \infty \right\},$$8 IDA-norm.
• For Bergman spaces, when $$L^p_\alpha = \left\{f \ \text{measurable} : \|f\|_{p,\alpha}^p = \int_{\mathbb{C}^n} |f(z)|^p e^{-\frac{\alpha p}{2}|z|^2} dv(z) < \infty \right\},$$9, $dv$0 and $dv$1; e.g., for $dv$2, $dv$3, $dv$4 is absolutely summing iff $dv$5.

7. Extensions and Further Remarks

All results for Fock spaces extend verbatim to general Fock-type spaces $dv$6 under the uniform convexity condition $dv$7 [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 $dv$8, $dv$9. 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 \in [1, \infty)$00-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).