Pettis integrability of functions with values in separable symmetrically-normed ideals and related norm estimates

Abstract: In this paper we will investigate Pettis integrability of $\mathcal{C}<sup>{\circ}_Φ(\mathcal{H})$-valued functions. We will study weakly$<sup>*$ integrable $\mathcal{B}(\mathcal{H})$-valued functions and establish sufficient conditions for such functions to be Pettis integrable as $\mathcal{C}<sup>{\circ}_Φ(\mathcal{H})$-valued functions. In addition, we prove the inequality $$\left|\int_E\mathscr{A}^{*\mathscr{B}dμ\right|_{Φ^{(p)}}} \leqslant |\mathscr{A}|<em>{L^{q_s}\cdot\left|\sqrt[p]{\int_E|\mathscr{B}|^pdμ}\right|}</em>{Φ<sup>{(p)}},$$ where $Φ<sup>{(p)}$ is $p$-modification of the function $Φ$ and the functions $\mathscr{A}$ and $\mathscr{B}$ belong to the suitable spaces of operator-valued functions. Finally, under some additional integrability assumptions on $\mathscr{B}$ we provide similar estimates of the Pettis norm.

• The paper presents sufficient conditions for Pettis integrability in separable s.n. ideals, extending integration theory for operator-valued functions.
• It introduces sharp Hölder-type inequalities and norm estimates, providing rigorous bounds for products and compositions of operator-valued functions.
• The paper reveals that Lᵖ spaces from operator-valued functions may lack vector space structure, prompting new approaches in noncommutative integration.

Summary of "Pettis integrability of functions with values in separable symmetrically-normed ideals and related norm estimates" (2607.02790)

Introduction and Background

The paper investigates the Pettis integrability of operator-valued functions, specifically those taking values in separable symmetrically-normed (s.n.) ideals within the algebra of bounded operators $B(H)$ on a separable Hilbert space $H$. The authors address both the foundational properties and nuanced integration theory for such functions, extending prior work on weak, Gelfand, and Bochner integrability for operator-valued mappings. These ideals, denoted by $C_\Phi(H)$ for a given s.n. function $\Phi$, generalize classical Schatten classes and capture a broad class of compact operator behaviors.

Operator Ideals and Preliminaries

The paper defines various families of operator ideals:

• $K(H)$ (finite-rank operators), $C_o(H)$ (compact operators), $B(H)$ (bounded operators), and $C_\Phi(H)$ where $\Phi$ is an s.n. function.
• The sequence spaces $c_0$, $H$0, and $H$1 are generalized via $H$2 to induce norms on singular values, governing the topology and duality of the corresponding operator ideals.

A detailed duality theory is reviewed, showing that every bounded linear functional on $H$3 arises from trace pairing with elements of the adjoint ideal $H$4, with norm control via an abstract Hölder inequality:

$H$5

for sequences $H$6, $H$7 in the respective spaces.

Integration of Operator-Valued Functions

The integration theory distinguishes between:

• Weak and weak$H$8 integrability: Scalarization against vectors or trace pairing, leading to notions such as Gelfand and weak$H$9 integrals.
• Strong and Bochner integrability: Norm-based integrability in the classical sense, but adapted for operator-valued functions with codomain $C_\Phi(H)$0.

A central focus is Pettis integrability: an operator-valued function $C_\Phi(H)$1 is Pettis integrable if, for every $C_\Phi(H)$2 in the $C_\Phi(H)$3-algebra, the vector $C_\Phi(H)$4 exists in $C_\Phi(H)$5 and coincides (via duality) with the integration of all scalarizations. For separable ideals, significant measurability simplifications enable a robust Pettis theory.

Main Results: Pettis Integrability and Norm Estimates

Sufficient Conditions for Pettis Integrability

The primary structural results provide sufficient conditions for the Pettis integrability of operator-valued functions with values in separable s.n. ideals. The results cover the passage from weak$C_\Phi(H)$6-integrability to Pettis integrability under countable additivity of the induced $C_\Phi(H)$7-valued measure and additional mild regularity and norm estimates.

Key theorems include:

• If $C_\Phi(H)$8 is weakly$C_\Phi(H)$9 integrable, $\Phi$0 for all $\Phi$1, and $\Phi$2 for all measurable $\Phi$3, then $\Phi$4 is $\Phi$5-Pettis integrable.
• For positive operator-valued functions and ideals of compact operators, Pettis integrability is achieved even without measure finiteness, provided the weak$\Phi$6 integral lies in $\Phi$7.

Hölder-Type Inequalities for Operator-Valued Functions

A second focus is on deriving sharp norm estimates and noncommutative Hölder-type inequalities for operator-valued functions:

• Generalizations of classical Hölder/Jensen inequalities are established for products and compositions of operator-valued mappings in various $\Phi$8-type spaces over $\Phi$9.
• Precise norm control is proven for products $K(H)$0, with estimates relating the norms of $K(H)$1 and $K(H)$2 in different $K(H)$3 contexts. For example, under certain $K(H)$4 conditions and suitable integrability, it is shown:

$K(H)$5

where $K(H)$6 is the $K(H)$7-modification of the s.n. ideal.

Non-Vector-Space Structure in Certain $K(H)$8 Spaces

The paper demonstrates by explicit construction that, except for $K(H)$9, the $C_o(H)$0-type spaces constructed for operator-valued functions over certain ideals generally fail to be vector spaces. That is, the sum of two $C_o(H)$1-integrable operator-valued functions need not be $C_o(H)$2-integrable, highlighting a sharp contrast with classical $C_o(H)$3 theory and reflecting noncommutativity and the operator norm's properties.

Jensen-Type Norm Inequalities

The authors also derive Jensen-type inequalities in this noncommutative operator context, providing upper bounds for the norm of the $C_o(H)$4-th moment (operator power) of a Pettis-integrable operator-valued function:

$C_o(H)$5

Implications and Future Directions

This work clarifies and extends foundational aspects of vector integration in the setting of operator algebras, particularly for separable s.n. ideals beyond the classical Schatten classes. The results have practical implications for noncommutative integration, vector measure theory, and quantum probability, where compact operator-valued functions arise naturally. The generalized Hölder-type inequalities and Pettis integrability criteria provide new tools for functional analysis and mathematical physics, especially in ergodic theory, noncommutative $C_o(H)$6 spaces, and quantum information.

From a theoretical perspective, the explicit construction of $C_o(H)$7-type spaces lacking vector space structure suggests rich avenues for future study of noncommutative integration, possibly inspiring the search for alternative, robust frameworks for operator-valued $C_o(H)$8 analysis and examining consequences in the theory of noncommutative Banach lattices and operator spaces.

Conclusion

The paper develops new sufficient criteria for Pettis integrability of functions valued in separable, symmetrically-normed operator ideals, establishes norm and product estimates via operator Hölder-type inequalities, and exposes limitations of conventional $C_o(H)$9 structures in the operator context. The findings deepen understanding of integration on operator spaces, opening further inquiry into the subtleties of noncommutative measure and functional analysis.