Gel'fand Integration of B(E, F*)-Valued Functions With Emphasis on (q, p)-Summing Operators

Abstract: We generalize results concerning Gel'fand integration of functions taking values in the space of operators on Hilbert spaces to certain Banach spaces. Building on ideas from \cite{M24} we provide sufficient conditions for the Gel'fand integral to be $(q,p)$-summing and we use the developed techniques to answer a question posed in the mentioned article. Applications to positive operator-valued functions between certain function spaces are also given.

• The paper establishes criteria under which the Gel'fand integral of operator-valued functions is (q, p)-summing.
• It introduces a bilinear approach linking tensor product continuity with (q, p)-summing operator properties in Banach spaces.
• The results resolve an open problem in symmetric norming ideals and offer improved estimates for positive operators on Banach lattices.

Gel'fand Integration of $(E, F^*)$-Valued Functions with Emphasis on $(q, p)$-Summing Operators

Introduction and Framework

The paper addresses the Gel'fand (weak$^*$) integration of operator-valued functions taking values in $(E, F^*)$, where $E$ and $F$ are complex Banach spaces. It extends established integration results from operators on Hilbert spaces to general Banach spaces and develops criteria under which operator-valued (o.v.) functions yield integrals that are $(q, p)$-summing. The paper not only adapts integration theory but also investigates the structure and behaviors of $(q, p)$-summing operators—maps satisfying an intricate summability condition connecting the weak and strong topologies of sequence spaces—and elucidates their interaction with Gel'fand integration. Applications to positive operator-valued functions between function spaces highlight practical cases and connections with Banach lattice theory.

Fundamentals of $(q, p)$-Summing Operators

A bounded linear operator $T: E \to F$ is said to be $(q, p)$0-summing if there exists $(q, p)$1 such that

$(q, p)$2

for any finite sequence $(q, p)$3 in $(q, p)$4. This property captures a transfer of summability from weakly $(q, p)$5-summable sequences in the domain to strongly $(q, p)$6-summable sequences in the codomain. The ideal $(q, p)$7 consists of all such operators, and the associated norm is denoted by $(q, p)$8. The $(q, p)$9-summing property is central to interpolation theory, operator ideals, and tensor product constructions in Banach space theory.

The paper establishes several key equivalences and duality relationships for $^*$0-summing operators. For instance, the formulation in terms of tensor product continuity—specifically, the continuity of

$^*$1

—connects the summing property with operator factorization through sequence and function spaces.

Gel'fand Integration of Operator-Valued Functions

Let $^*$2 be a measure space. An o.v. function $^*$3 is weak$^*$4-measurable if for all $^*$5, $^*$6, the scalar function $^*$7 is measurable; it is integrable if these functions are integrable. The Gel'fand integral of $^*$8 over $^*$9 is the unique element of $(E, F^*)$0 defined by

$(E, F^*)$1

This extends the notion of weakly integrable vector-valued functions to the case of operator-valued functions, leveraging the duality and tensor product structure of Banach spaces.

The characterization of weak$(E, F^*)$2 measurability and integrability for operator-valued mappings is given in a bilinear (tensorial) form, which simplifies verification and interrelates with tensor norm topology.

Sufficient Conditions for $(E, F^*)$3-Summability of Gel'fand Integrals

A principal contribution of the paper is providing sufficient conditions for the Gel'fand integral of an o.v. function to be $(E, F^*)$4-summing. The main theorem asserts that if $(E, F^*)$5 is weak$(E, F^*)$6-measurable and, for all $(E, F^*)$7 and $(E, F^*)$8,

$(E, F^*)$9

then for all $E$0,

$E$1

When $E$2, improved estimates are available due to the availability of a Banach predual for the ideal $E$3. In the positive case—when all $E$4 are positive operators between Banach lattices—these summability properties can be deduced under even weaker integrability conditions and are closely connected to the structure of function spaces such as $E$5 and $E$6.

Solution to an Open Problem on Symmetric Norming Ideals

The paper addresses a previously unresolved question related to integrability in the context of symmetric norming (s.n.) ideals of operators on Hilbert space. Specifically, the supremum

$E$7

over all orthonormal systems $E$8 is shown to be finite provided the corresponding pointwise integrals are finite. This result leverages an additive decomposition of weakly $E$9-summable sequences in a Hilbert space (into combinations of orthonormal sequences), the functional properties of s.n. functions, and the Baire category theorem. It establishes quantitative uniformity in the Gel'fand integrability condition for a large class of operator ideals.

Special Classes: Positive Operators and Banach Lattices

A notable set of examples and counterexamples demonstrate the tightness of the assumptions. If $F$0 is positive and weak$F$1-integrable, then for each $F$2, the Gel'fand integral is positive and $F$3-summing. Conversely, without positivity, the existence of non-$F$4-summing operators as Gel'fand integrals of pointwise $F$5-summing maps is shown.

Implications and Prospective Developments

The findings robustly generalize earlier results on weak integration of operator-valued functions. The explicit bilinear criteria for summability, accompanied by operator norm estimates, facilitate the extension of integration theory from Hilbert spaces to Banach space settings, including operator ideals and function space morphisms. In the context of functional analysis, this underpins the treatment of vector martingales, stochastic integrals, and operator-valued measures.

For operator theory, the results refine our understanding of the behavior of operator ideals under integration, especially in Banach lattice and symmetric ideal environments. Further, they yield improved control over the summing norms of integrals, which is foundational for interpolation, Grothendieck-type inequalities, and tensor product theories.

The resolution of the uniform bound for symmetric norming ideals points toward deep connections with Banach space geometry and suggests avenues for subsequent research into vector measure integration, duality for operator ideals, and Banach lattice theory.

Conclusion

The paper provides a comprehensive advancement in the Gel'fand integration theory of $F$6-valued functions, generalizing summability criteria to Banach spaces and addressing subtle questions regarding operator ideals and symmetric norms. The results are accompanied by detailed functional-analytic techniques, sharp norm estimates, and illustrative examples. This establishes a refined theoretical foundation for integration in operator-valued and Banach space frameworks, with direct consequences for advancing operator theory, tensor products, and applications where operator-valued measures arise.