Lipschitz Numerical Index in Banach Spaces

Updated 7 July 2026

Lipschitz numerical index is a nonlinear analogue of the classical numerical index, defined via the numerical radius of Lipschitz self-maps fixing zero.
It incorporates structural results like the Lumer’s formula analogue and criteria for index one, linking perturbation techniques to Banach space geometry.
The index bridges classical and Lipschitz operator theory through differentiability and free-space linearization while highlighting open problems in both real and complex settings.

The Lipschitz numerical index of a Banach space $X$ is the nonlinear analogue of the classical numerical index obtained by replacing bounded linear operators with Lipschitz self-maps fixing $0$. It is defined through the numerical radius of Lipschitz operators and measures how effectively the Lipschitz numerical radius controls the Lipschitz norm on the algebra $\operatorname{Lip}_0(X)$ . A recurring theme in the literature is that this nonlinear invariant often agrees with the classical numerical index $n(X)$ , even though $\operatorname{Lip}_0(X)$ is much larger than $L(X)$ ; at the same time, later work shows that the coincidence is not automatic in the complex setting (Wang et al., 2012, Pérez-Hernández, 24 Jul 2025).

1. Foundational definitions

Let $X,Y$ be Banach spaces over $\mathbb{R}$ or $\mathbb{C}$ . A mapping $T:X\to Y$ is Lipschitz if there exists $0$0 such that

$0$1

The space $0$2 consists of all Lipschitz maps sending $0$3 to $0$4, equipped with the Lipschitz norm

$0$5

For $0$6, the norming set

$0$7

collects the functionals that norm $0$8. For $0$9, the numerical range is defined by

$\operatorname{Lip}_0(X)$ 0

and the numerical radius is

$\operatorname{Lip}_0(X)$ 1

The Lipschitz numerical index is then

$\operatorname{Lip}_0(X)$ 2

equivalently,

$\operatorname{Lip}_0(X)$ 3

If $\operatorname{Lip}_0(X)$ 4 is linear and bounded, this recovers the classical numerical radius, and therefore

$\operatorname{Lip}_0(X)$ 5

The literature uses several notational conventions for the Lipschitz numerical radius, including $\operatorname{Lip}_0(X)$ 6, $\operatorname{Lip}_0(X)$ 7, and $\operatorname{Lip}_0(X)$ 8 (Wang et al., 2012, Choi et al., 2021, Pérez-Hernández, 24 Jul 2025).

A useful state-space formulation is

$\operatorname{Lip}_0(X)$ 9

In this notation,

$n(X)$ 0

which is the form used in later developments on renorming, attainment, and function-space constructions (Choi et al., 2021).

2. Numerical-radius formulas and the case $n(X)$ 1

A central structural result is the Lipschitz analogue of Lumer’s formula. For every Banach space $n(X)$ 2 and every $n(X)$ 3,

$n(X)$ 4

and therefore

$n(X)$ 5

This identifies the numerical radius with the first-order growth of the Lipschitz norm of perturbations $n(X)$ 6. A dense-set version is also available: if

$n(X)$ 7

and $n(X)$ 8 projects densely onto $n(X)$ 9, then

$\operatorname{Lip}_0(X)$ 0

These formulas make the numerical radius computable from dense configurations of pairs and supporting functionals (Wang et al., 2012).

The same perturbative framework yields an exact criterion for numerical index $\operatorname{Lip}_0(X)$ 1. For $\operatorname{Lip}_0(X)$ 2,

$\operatorname{Lip}_0(X)$ 3

and

$\operatorname{Lip}_0(X)$ 4

Consequently,

$\operatorname{Lip}_0(X)$ 5

This is the Lipschitz counterpart of the alternative Daugavet equation and provides the basic operator-theoretic test used throughout the subject (Wang et al., 2012).

3. Relation with the classical numerical index

The earliest identification results use differentiability. If $\operatorname{Lip}_0(X)$ 6 is Gâteaux differentiable everywhere, then $\operatorname{Lip}_0(X)$ 7 for all $\operatorname{Lip}_0(X)$ 8, and in the scalar-invariant setting one has

$\operatorname{Lip}_0(X)$ 9

Using the theorem that every Lipschitz map from an open set $L(X)$ 0 into $L(X)$ 1 is Gâteaux differentiable outside an Arónszajn null set when $L(X)$ 2 is separable and $L(X)$ 3 has the Radon–Nikodým property, it follows that if $L(X)$ 4 is a separable Banach space with the Radon–Nikodým property, then

$L(X)$ 5

Among the consequences recorded are

$L(X)$ 6

and, if $L(X)$ 7 and $L(X)$ 8 is separable,

$L(X)$ 9

The same work also states, in the complex case,

$X,Y$ 0

These results show that, on large classes of spaces, the nonlinear invariant is recovered from linear derivatives (Wang et al., 2012).

Later work extends the equality $X,Y$ 1 in a different direction. For real Banach spaces, it proves that

$X,Y$ 2

whenever $X,Y$ 3 is separable, and also that

$X,Y$ 4

for every real Banach space $X,Y$ 5. The proof mechanisms are distinct from the RNP argument: separable spaces are treated by Gaussian convolution and weak Gâteaux differentiability, while dual spaces are treated by invariant means and a linearization-by-translation method. The same source explicitly states that the positive results are real-space results and gives a complex counterexample: $X,Y$ 6 This produces a strict separation between the Lipschitz and classical indices in the complex setting and reframes the open problem as a real-space question rather than a fully general one (Pérez-Hernández, 24 Jul 2025).

The sources are not fully uniform on the complex theory. One source states the general complex lower bound

$X,Y$ 7

for every complex Banach space, while the later source gives $X,Y$ 8. This suggests that the complex case requires careful attention to the precise framework and conventions being used (Wang et al., 2012, Pérez-Hernández, 24 Jul 2025).

4. Spaces and constructions with explicit values or transfer formulas

A substantial part of the theory is devoted to identifying classes of spaces for which $X,Y$ 9 and to proving stability under standard Banach-space constructions. The following exact results are established.

Construction or class	Statement	Source
Real lush space $\mathbb{R}$ 0	$\mathbb{R}$ 1	(Wang et al., 2012)
$\mathbb{R}$ 2-rich subspace $\mathbb{R}$ 3	$\mathbb{R}$ 4	(Wang et al., 2012)
Compact Hausdorff $\mathbb{R}$ 5	$\mathbb{R}$ 6	(Wang et al., 2012)
Positive measure space $\mathbb{R}$ 7	$\mathbb{R}$ 8	(Wang et al., 2012)
$\mathbb{R}$ 9-finite $\mathbb{C}$ 0	$\mathbb{C}$ 1	(Wang et al., 2012)
Compact Hausdorff $\mathbb{C}$ 2	$\mathbb{C}$ 3	(Wang et al., 2012)
$\mathbb{C}$ 4-, $\mathbb{C}$ 5-, $\mathbb{C}$ 6-sums	$\mathbb{C}$ 7	(Wang et al., 2012)

For the classical scalar examples, these formulas imply

$\mathbb{C}$ 8

and also

$\mathbb{C}$ 9

The proofs use several different mechanisms: direct estimates for $T:X\to Y$ 0, lushness and join-hull arguments for real lush spaces and $T:X\to Y$ 1-rich subspaces, and finite-rank conditional expectation or averaging constructions for $T:X\to Y$ 2- and $T:X\to Y$ 3-spaces (Wang et al., 2012).

Later work broadens the transfer theory. For vector-valued function spaces, if $T:X\to Y$ 4 is norming and pointwise composition $T:X\to Y$ 5 preserves the relevant algebra, then

$T:X\to Y$ 6

In particular,

$T:X\to Y$ 7

For Köthe-Bochner spaces $T:X\to Y$ 8, the established general inequality is

$T:X\to Y$ 9

hence, for $0$00,

$0$01

For absolute sums,

$0$02

under the hypotheses stated for the absolute norm $0$03. Finally, if $0$04 contains a dense union of an increasing family of one-complemented subspaces $0$05, then

$0$06

Combining these principles yields the formulas

$0$07

and, whenever $0$08 is infinite-dimensional,

$0$09

These results show that $0$10 is compatible with a broad range of geometric and function-space constructions, although for absolute sums only the upper estimate is obtained in general (Choi et al., 2021).

5. Renorming, numerical-radius attainment, and Lipschitz Daugavet centers

The Lipschitz numerical index admits a renorming theory parallel to the classical one. Let $0$11 denote the set of equivalent norms on $0$12, endowed with

$0$13

For a norm $0$14, one defines the corresponding state set $0$15, the numerical radius $0$16, and the renormed Lipschitz numerical index $0$17. The map

$0$18

is uniformly continuous on each bounded subset of $0$19, and

$0$20

is continuous on $0$21. Consequently, the renorming range

$0$22

is an interval. The precise result is continuity plus the interval property; no full classification of the possible interval is given (Choi et al., 2021).

A second line of development concerns Lipschitz numerical-radius attaining maps. A map $0$23 attains its Lipschitz numerical radius if there exists $0$24 such that

$0$25

These maps exhibit a rigid segment structure: if $0$26 attains its Lipschitz numerical radius at $0$27, then for every $0$28, with $0$29, the same map attains at

$0$30

The main global conclusion is negative:

for every Banach space $0$31, the set of Lipschitz numerical-radius attaining maps is not dense in $0$32. This is one of the strongest universal obstructions in the theory. There is also a derivative transfer principle: if $0$33 attains its Lipschitz numerical radius at $0$34 and is Gâteaux differentiable on $0$35, then every derivative $0$36 with $0$37 attains the classical numerical radius at

$0$38

Thus Lipschitz numerical-radius attainment, when present, forces linear numerical-radius attainment for the derivatives (Choi et al., 2021).

The same paper introduces Lipschitz Daugavet centers. A map $0$39 is a Lipschitz Daugavet center if

$0$40

for every rank-one Lipschitz map $0$41. Using slope sets

$0$42

and Lip-slices, it is shown that if $0$43 is a Daugavet center, then

$0$44

for every $0$45 whose slope set is an SCD-set, and more generally whenever $0$46 has the Radon–Nikodým property, the Asplund property, the convex point of continuity property, or absence of $0$47-sequences. In particular, for linear maps,

$0$48

This places the numerical-index theory in close contact with Daugavet-type geometry (Choi et al., 2021).

An important conceptual reformulation identifies the Lipschitz numerical index with a linear numerical index relative to a distinguished operator. Given Banach spaces $0$49 and a norm-one operator $0$50, the numerical index with respect to $0$51 is

$0$52

equivalently the greatest constant $0$53 such that

$0$54

For a Banach space $0$55, let $0$56 be its Lipschitz-free space, let $0$57 be the canonical embedding, and let $0$58 be the linearization of a Lipschitz map $0$59. The identity map on $0$60 linearizes to a norm-one operator

$0$61

The bridge theorem states that

$0$62

and therefore

$0$63

This shows that the nonlinear theory is exactly an instance of the numerical index with respect to an operator, with the canonical reference operator $0$64 supplied by Lipschitz-free linearization (Kadets et al., 2019).

A related, but distinct, development studies the ordinary numerical index of Lipschitz-free spaces. For a three-point metric space $0$65, the space $0$66 is $0$67-dimensional, and its numerical index admits an explicit metric formula. If the three points are aligned, then

$0$68

if they form a nondegenerate triangle, the value is given by a formula involving Gromov products and a second metric term, and the set of all possible values is exactly

$0$69

Moreover, every value in this interval is attained by $0$70-dimensional subspaces of $0$71 whenever $0$72 with $0$73 has non-empty interior. That work explicitly notes that it studies the ordinary numerical index of $0$74, not a separate Lipschitz numerical index of nonlinear maps; its relevance here is the duality

$0$75

which places free-space geometry adjacent to the Lipschitz numerical-index program (Cobollo et al., 2023).

Several open directions remain. One source ends with the question

$0$76

Later work answers this in the complex setting by the example $0$77, but retains the real-space version as an open problem: does

$0$78

hold for every real Banach space $0$79? Other questions concern estimates in terms of property $0$80, equality for vector-valued disk algebras $0$81 and $0$82, the weak and weak* function spaces $0$83 and $0$84, and whether the reverse inequality for absolute sums holds under suitable assumptions (Wang et al., 2012, Pérez-Hernández, 24 Jul 2025, Choi et al., 2021).

In this form, the Lipschitz numerical index sits at the intersection of nonlinear operator theory, Banach-space geometry, Daugavet-type equations, and Lipschitz-free linearization. Its defining question is whether the passage from $0$85 to $0$86 creates genuinely new numerical-index phenomena. The known theory shows both rigidity and variation: extensive families of spaces satisfy $0$87, many classical spaces have $0$88, renorming yields interval-valued ranges, numerical-radius attainment is universally non-dense, and the complex case already exhibits strict separation.