Domination Principle for Positive Operators

• The domination principle for positive operators is the transfer of properties from a dominating operator to a dominated one under specific order conditions.
• It is applied to establish the inheritance of compactness, analyticity, and sparse domination in diverse settings such as Banach lattices and non-commutative algebras.
• The framework utilizes ordered Banach spaces, kernel representations, and spectral projections to reveal structural insights in functional, harmonic, and ergodic analyses.

The domination principle for positive operators concerns structural inequalities, hereditary ideal properties, and transfer of regularity within ordered (often partially ordered) algebraic and analytical contexts. Such principles underlie much of functional analysis, operator algebra, ergodic theory, and harmonic analysis, asserting that if a positive operator $T$ is order-bounded by another positive operator $S$ having a given property (e.g., compactness, summability, analyticity), then $T$ inherits this property under qualitative assumptions on the ambient structure or the operators involved. Modern formulations appear in both commutative and non-commutative analysis, semi-group and spectral theory, Banach lattices, multilinear environments, Dirichlet forms, and stochastic analysis.

1. Formal Statement of Domination

Let $E, F$ be ordered normed spaces (e.g., Banach lattices, $C^*$-algebras, preduals of von Neumann algebras), and let $T, S: E \to F$ be positive operators. One says that $T$ is dominated by $S$ (notation: $0 \le T \le S$) if

$$0 \leq T(x) \leq S(x) \quad \text{for all } x \in E_+.$$

In probabilistic or dynamical contexts, domination may refer to (eventual or asymptotic) inequalities involving time-parameterized operator families; for example, semigroups $(T(t))_{t\ge0}, (S(t))_{t\ge0}$ satisfy $T(t) \leq S(t)$ for all $t$ if $S$ dominates $T$ (Arora et al., 2022).

In multilinear, polynomial, or tensorial settings, domination is formulated via inequalities controlling the joint action of the operator on positive elements and sequences in Banach lattices, with appropriate norms and summability conditions (see, e.g., (Belaada et al., 10 Sep 2025, Ferradi et al., 1 Feb 2024)).

When positive differential operators or Dirichlet forms are involved, domination connects to inequalities between quadratic forms or resolvents associated to operators, often interpreted as domination between Markov processes or killed processes (Li et al., 12 Dec 2024).

2. Inheritance of Regularity and Ideals

A key theme is the transfer of membership in operator ideals under domination:

• Compactness: If $S$ is compact and $0 \leq T \leq S$, then under suitable geometric or order-theoretic assumptions (e.g., $E$ scattered or $F$ compact for $C^*$-algebras), $T$ is also compact ((Oikhberg et al., 2012), Theorem 2.2.1). For pre-Riesz spaces, analogous statements hold using Riesz completions and extension of order continuous norms (Gaans et al., 2018).
• Weak/Dunford–Pettis Compactness: If $T$ is Dunford–Pettis and $F$ is o-Dedekind complete, $S$ dominated by $T$ is also weak* Dunford–Pettis (Chen et al., 2013).
• Sparse Domination: For (possibly nonlocal or nonoperator) functions or forms in harmonic analysis, positive operators can be pointwise dominated by positive "sparse forms," i.e., local averages over carefully selected cube families, enabling the transfer of weighted and vector-valued norm inequalities (Lerner et al., 2021, Plinio et al., 2016, Culiuc et al., 2016).
• Analyticity: If $T$ is a positive, analytic $C_0$-semigroup and $0 \leq S(t) \leq T(t)$ for all $t$, then $S$ is analytic (solving a problem posed by Arendt in 2004) (Glück, 2022).
• Multilinear Domination: The class of positive weakly $(q, r)$-dominated multilinear operators forms a positive multi-ideal closed under composition and admits tensorial representations mirroring classical Pietsch-type domination and factorization theorems (Belaada et al., 10 Sep 2025, Ferradi et al., 1 Feb 2024).

3. Structural and Technical Frameworks

Ordered Banach and Function Spaces

The analysis typically proceeds within the context of ordered Banach spaces (OBS) or Banach lattices. The geometric/convex structure (e.g., order intervals defined as $[0, a]$) is often crucial. In non-commutative settings, fully symmetric spaces associated to traces on von Neumann algebras are foundational (Oikhberg et al., 2012).

Key properties:

• Order Completeness / Dedekind Completeness: Essential for duality and for ensuring every order-bounded family has a supremum (or infimum), underpinning approximation and decomposition arguments.
• Order Continuous Norms: Necessary for the transfer of compactness under domination in both commutative and non-commutative settings.

Boolean Algebras, Up/Down Decomposition, and Fragments

Positive operators (especially in vector lattices and their generalizations, such as Uryson operators) admit fragmentations into Boolean algebras of "order-disjoint" components, enabling fine-grained structural decompositions ("up-and-down theorem") (Pliev, 2015).

Spectral Projection and Perron-Frobenius/Kreĭn–Rutman Theory

Domination conditions play a foundational role in spectral theory:

• If resolvent or semigroup projections satisfy $P f \succcurlyeq u$ for positive $f$ and quasi-interior points $u$, strong positivity and simplicity of leading eigenvalues follow (i.e., Perron–Frobenius theorems) (Daners et al., 2017, Arora et al., 2022).
• Kreĭn–Rutman type results derive positive eigenvectors/eigenfunctionals under domination or eventual positivity (Daners et al., 2017).

Dirichlet Forms, Probabilistic Killing, and Robin-Type Operators

Via Ouhabaz's criterion, domination of semigroups is equivalent to domination of associated Dirichlet forms, with connections to stochastic processes. Killing transformations by multiplicative functionals produce subordinate forms, with extra terms expressible through bivariate Revuz measures. Operators sandwiched between Dirichlet and Neumann Laplacians correspond precisely (in both local and nonlocal Robin-boundary cases) to Dirichlet forms with explicit boundary measures (Li et al., 12 Dec 2024, Akhlil, 2020).

4. Domination in Non-Commutative, Multilinear, and Harmonic Analysis

• Non-Commutative Settings: Compactness and other operator ideal properties are inherited provided the domain or codomain has appropriate order/topological properties (e.g., a scattered $C^*$-algebra or a compact algebra) (Oikhberg et al., 2012).
• Multilinear/Sparse Domination: Multilinear forms arising in singular integral theory (e.g., the bilinear Hilbert transform or variational Carleson operator) are sharply dominated by positive sparse forms, which immediately yield norm and weighted norm estimates (Culiuc et al., 2016, Plinio et al., 2016).
• Operator-Free Approaches: Sparse domination principles extend beyond classical operators; for families of functions satisfying certain "chain rule" properties, domination by sparse forms holds even for objects not directly localizable as operators (applications include generalized Poincaré–Sobolev inequalities and tent spaces) (Lerner et al., 2021).

5. Analytic and Long-Time Properties via Domination

• Eventual Domination: If one positive $C_0$-semigroup (or its resolvent) eventually dominates another (e.g., $|e^{tA}f| \le e^{tB}|f|$ for large $t$), then strong asymptotic and qualitative features—such as strong convergence, spectral simplicity, and ergodic limit theorems—can be transferred (Arora et al., 2022, Glück et al., 2018).
• Lower Bounds and Strong Convergence: Existence of a fixed positive element asymptotically dominated by all semigroup orbits implies strong convergence to rank-one projections under mild structural assumptions (Glück et al., 2018).

6. Limitations, Counterexamples, and Optimality

• Non-Heredity of Stronger Positivity: Properties such as $2$-positivity, complete positivity, or decomposability are not necessarily inherited by operators dominated by positive ones, even if the dominating operator is of rank one or highly regular (Oikhberg et al., 2012).
• Need for Structural Hypotheses: Without suitable order completeness (e.g., $\sigma$-Dedekind completeness, quasi-interior points, order unit norms), domination may fail to deliver compactness or other desirable properties (Chen et al., 2013, Gaans et al., 2018).
• Sharpness: In Banach lattices, order unit and pervasive structures are essential for the extension of domination principles in more general pre-Riesz spaces (Gaans et al., 2018).

7. Extended Frameworks and Applications

• Kernel Operators and Representation: A positive operator between $L^p$-spaces admits a kernel representation if and only if it smooths positive functions to those with lower semicontinuous representatives for a suitable topology. In Banach lattices, this is equivalent to the positive range being representable over a countable set of positive vectors; domination by a nontrivial kernel operator has a parallel characterization (Gerlach et al., 2022).
• Discrete and Categorical Domination: Expansive, monotone, or closure operators on finite set systems embody a discrete domination framework; Galois connections and categorical pullbacks characterize more complex structural relations, including antimatroid (uniquely generated) closure operators (Pfaltz, 2015).

8. Representative Results and Theorems

Setting	Domination Condition	Inheritance/Consequence
Banach lattice	$0 \le T \le S$, $S$ is compact	$T$ is compact if domain is scattered or codomain is compact (Oikhberg et al., 2012)
Banach lattice	$0 \le S \le T$, $T$ weak* Dunford–Pettis	$S$ is weak* Dunford–Pettis if $F$ is o-Dedekind complete (Chen et al., 2013)
Uryson operators	$0 \le S \le T$ (T order narrow)	$S$ is also order narrow (Pliev, 2015)
Dirichlet forms	$(\mathcal{E}', D(\mathcal{E}'))$ dominated by $(\mathcal{E}, D(\mathcal{E}))$	Induced by killing; sandwiched semigroups = Robin-type forms (Li et al., 12 Dec 2024)
Positive semigroups	$0 \le S(t) \le T(t)$, $T$ analytic	$S$ analytic (Glück, 2022)
Multilinear operator	Positive weakly $(q, r)$-dominated	Pietsch domination, tensorial dual representation (Belaada et al., 10 Sep 2025)

9. Open Problems and Future Directions

• Non-symmetric and Non-local Forms: Comprehensive understanding of domination among non-symmetric Dirichlet forms, especially in non-local Robin-type settings, is now accessible and characterized by explicit bivariate measures, resolving prior open problems (Li et al., 12 Dec 2024).
• Extension to Non-positive or Nonlinear Operators: Whether analytic inheritance holds for non-positive semigroups under domination, and similar properties for nonlinear or more generalized operator classes, is under continued investigation (Glück, 2022).
• Complete Hereditary Descriptions: There is ongoing interest in fully characterizing which operator ideals or regularity properties (beyond positivity and compactness) are preserved under domination—especially in operator systems and non-commutative set-ups (Oikhberg et al., 2012).

10. Summary

Domination principles for positive operators unify and generalize hereditary transfer of compactness, regularity, analyticity, and summability properties in the presence of order and cone structures, with precise boundaries determined by the interplay between algebraic, geometric, and order-theoretic features of the ambient space. The theory now encompasses non-commutative function spaces, multilinear and polynomial operators, Dirichlet forms (including probabilistic killing and Robin boundary conditions), and modern harmonic analysis via sparse domination. Recent work provides both abstract characterizations (e.g., via kernel representations, tensor norms, fragmentation, and categorical pullbacks) and concrete operator-theoretic results, forming a core element in current research in functional analysis, operator theory, and mathematical physics.