Rank-One Multiplicative Perturbation

• The paper shows that adding a rank-one operator can increase the norm of an operator in reflexive Banach spaces without ensuring norm attainment.
• It distinguishes the V-property, Weak Maximizing Property, and Compact Perturbation Property through explicit constructions and counterexamples.
• The findings challenge classical intuitions in operator theory and motivate further research into the stability of norm attainment under minimal perturbations.

A rank-one multiplicative perturbation refers to the modification of an operator—typically within the context of a Banach or Hilbert space—by a rank-one operator using an additive or multiplicative structure, with significant implications for the geometry of operator norms, the norm-attaining property, and spectral theory. This article focuses on rank-one operators as multiplicative perturbations in the sense of norm-attaining phenomena for operators between Banach spaces, following the framework and results of (Martínez-Cervantes et al., 2023).

1. Definitions and Fundamental Concepts

A rank-one operator $R \in \mathcal{L}(X,Y)$ (where $X$, $Y$ are Banach spaces) is one whose range is one-dimensional. Explicitly, $R$ can be written as

$R(x) = f(x)\, y \quad \text{for all } x \in X,$

where $f\in X^*$ (the dual of $X$) and $y \in Y\setminus\{0\}$. Such operators represent the simplest nontrivial nonzero modifications in linear operator theory.

An operator $T\in\mathcal{L}(X,Y)$ attains its norm if there exists $x\in B_X$ (the closed unit ball of $X$) such that $\|T(x)\|=\|T\|$. Understanding how this property behaves under small (in norm) or structured (such as rank-one) perturbations is a classical and delicate issue in the geometry of Banach spaces and operator theory.

2. Existence of Non-Norm-Attaining Rank-One Perturbations

A principal result established in (Martínez-Cervantes et al., 2023) is that for any infinite-dimensional reflexive Banach space $X$, there exists a reflexive Banach space $Y$ and operators $T, R \in \mathcal{L}(X, Y)$ with $R$ a rank-one operator such that

$\|T+R\|>\|T\|,$

yet $T+R$ fails to attain its norm. This result demonstrates that the addition of a rank-one operator can strictly increase the norm of $T$, but it does not guarantee the existence of a maximizing vector; norm attainment can indeed fail.

This construction answers a question posed in earlier work by S. Dantas and the first two authors, which asked whether every norm increase via a compact (or even rank-one) perturbation must yield a norm-attaining operator in reflexive Banach spaces. The negative resolution underscores the subtleties of norm attainment even for minimal-rank modifications.

3. Relationships Among the V-Property, Weak Maximizing Property, and CPP

The broader context involves several structural properties relating to norm-attainment:

• V-Property: A pair $(X,Y)$ has the V-property if for every $T\in\mathcal{L}(X,Y)$, there exists a norm-one operator $S\in\mathcal{L}(Y,X)$ with the spectral radius $r(TS)=\|T\|$. When $T$ has certain geometric features (e.g., a strictly singular hump), this condition is equivalent to norm attainment.
• Weak Maximizing Property (WMP): $(X,Y)$ has the WMP if every operator with a maximizing sequence which is not weakly null attains its norm.
• Compact Perturbation Property (CPP): Any compact perturbation $K$ added to $T$ with $\|T+K\|>\|T\|$ produces a norm-attaining operator.

Both the V-property and WMP imply the CPP, but they are not equivalent; there exist pairs with the WMP but not the V-property, and vice versa. This separation is highlighted by examples and counterexamples discussed in Section 3 of (Martínez-Cervantes et al., 2023). Understanding these properties and their (non-)equivalence is crucial for assessing the stability of the norm-attainment phenomenon under perturbations.

4. Mathematical Formulations and Mechanisms

The essential constructions revolve around the following formulations:

• Rank-one operator representation:

$R(x) = f(x)\, y,$

with $f\in X^*$, $y\in Y$.

• Norm attainment equation:

$\text{$T$ attains its norm} \iff \exists\,x\in B_X : \|T(x)\| = \|T\|$

• Perturbation norm inequality:

$\|T+R\| > \|T\| \quad \text{and $T+R$ does not attain its norm},$

showing that even minimal rank adjustments can disrupt norm-attaining status.

• Operator constructed into a direct sum: For certain constructed Banach space $Z$,

$T+R: X \to Z,$

is assembled to exemplify the failure of the CPP.

These mechanisms underpin the constructed counterexamples and provide a template for the analysis of related operator-theoretic questions.

5. Applications and Theoretical Consequences

The study of rank-one multiplicative perturbations in norm-attaining theory has several notable implications:

• Operator Theory and Optimization: Understanding when operators attain their norm is relevant for stability and optimality in numerical linear algebra, optimization, and variational analysis.
• Geometry of Banach Spaces: The existence of non-norm-attaining rank-one perturbations in reflexive Banach spaces challenges classical intuitions. The result illustrates that reflexivity does not guarantee the stability of the norm-attaining property under rank-one modifications.
• Further Directions in Perturbation Theory: The failure of the CPP in this context leads to several open research problems:
  • Whether similar failures can occur for other low-rank or more general classes of perturbations.
  • Full characterization of reflexive Banach spaces where all operators attain their norm.
  • Investigation of isomorphic invariance phenomena: notably, the CPP is not preserved under isomorphism, as demonstrated by differences between $(X, c)$ and $(X, c_0)$ for reflexive $X$.
• Interplay with the V-property and WMP: Exploring whether one of these properties implies the other, and further examining the structural reasons for their distinction, remain open, as explicit counterexamples illustrate their independence.

6. Summary Table: Properties and Implications

Property	Definition	Implies
V-property	$T$ is a V-operator if $\exists$ norm-one $S$ with $r(TS)=\|T\|$	CPP
Weak Maximizing Prop.	Any maximizing sequence not weakly null → norm attainment	CPP
Compact Perturbation	Compact $K$: $\|T+K\|>\|T\|$ implies norm attainment	—

The table encapsulates the logical dependencies discussed; counterexamples show no equivalence between V-property and WMP, even though both imply CPP.

7. Conclusion

Rank-one multiplicative perturbations, despite their simplicity, play a nuanced and powerful role in the theory of norm-attaining operators on Banach spaces. The explicit constructions in (Martínez-Cervantes et al., 2023) demonstrate that the addition of a rank-one operator can raise the operator norm without producing norm attainment, even in the context of reflexive Banach spaces. This outcome not only answers long-standing questions but uncovers the subtle relationships among geometric properties (V-property, WMP, CPP) in operator theory. The work motivates further study into the interplay between operator structure, perturbations, and the attainable geometry of Banach spaces.