Daugavet Centers
An operator G: X → Y is said to be a Daugavet center if ||G + T|| = ||G|| + ||T|| for every rank-1 operator T: X → Y . The main result of the paper is: if G: X →! Y is a Daugavet center, Y is a subspace of a Banach space E, and J : Y → E is the natural embedding operator, then E can be equivalently...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2010
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| Cite this: | Daugavet Centers / T. Bosenko, V. Kadets // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 1. — С. 3-20. — Бібліогр.: 14 назв. — англ. |
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| citation_txt | Daugavet Centers / T. Bosenko, V. Kadets // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 1. — С. 3-20. — Бібліогр.: 14 назв. — англ. |
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| description | An operator G: X → Y is said to be a Daugavet center if ||G + T|| = ||G|| + ||T|| for every rank-1 operator T: X → Y . The main result of the paper is: if G: X →! Y is a Daugavet center, Y is a subspace of a Banach space E, and J : Y → E is the natural embedding operator, then E can be equivalently renormed in such a way that J ○ G : X → E is also a Daugavet center. This result was previously known for the particular case X = Y, G = Id and only in separable spaces. The proof of our generalization is based on an idea completely di®erent from the original one. We also give some geometric characterizations of the Daugavet centers, present a number of examples, and generalize (mostly in straightforward manner) to Daugavet centers some results known previously for spaces with the Daugavet property.
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Journal of Mathematical Physics, Analysis, Geometry
2010, vol. 6, No. 1, pp. 3–20
Daugavet Centers
T. Bosenko and V. Kadets∗
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv, 61077, Ukraine
E-mail:t.bosenko@mail.ru
vova1kadets@yahoo.com
Received February 19, 2009
An operator G: X → Y is said to be a Daugavet center if ‖G + T‖ =
‖G‖ + ‖T‖ for every rank-1 operator T : X → Y . The main result of the
paper is: if G: X → Y is a Daugavet center, Y is a subspace of a Banach
space E, and J : Y → E is the natural embedding operator, then E can be
equivalently renormed in such a way that J ◦G : X → E is also a Daugavet
center. This result was previously known for the particular case X = Y ,
G = Id and only in separable spaces. The proof of our generalization is
based on an idea completely different from the original one. We also give
some geometric characterizations of the Daugavet centers, present a number
of examples, and generalize (mostly in straightforward manner) to Daugavet
centers some results known previously for spaces with the Daugavet prop-
erty.
Key words: Daugavet center, Daugavet property, renorming.
Mathematics Subject Classification 2000: 46B04 (primary); 46B03,
46B25, 47B38 (secondary).
1. Introduction
A Banach space X is said to have the Daugavet property if all the operators
T : X → X of rank-1 satisfy the Daugavet equation
‖Id + T‖ = 1 + ‖T‖. (1.1)
Several classical spaces have the Daugavet property: C(K), where K is perfect
[1], L1(µ), where µ has no atoms [2], and certain functional algebras such as the
disk algebra A(D) or the algebra of bounded analytic functions H∞ ([12, 14]).
∗Research of the second named author was conducted during his stay in the University of
Granada and was supported by Junta de Andalućıa grant P06-FQM-01438.
c© T. Bosenko and V. Kadets, 2010
T. Bosenko and V. Kadets
Geometric and linear-topological properties of such spaces were studied in-
tensively during the last two decades (see the survey paper [13] and most recent
developments in [8, 3, 4]). In particular, if X is a space with the Daugavet pro-
perty, then every weakly compact operator, even every strong Radon–Nikodým
operator on X, and every operator on X not fixing a copy of `1, fulfill (1.1) as
well ([6, 11]). These spaces contain subspaces isomorphic to `1, cannot have the
Radon–Nikodým property, never have an unconditional basis and even never em-
bed into a space having an unconditional basis. The key to the later embedding
property is the following theorem:
Theorem 1.1. [6, Th. 2.5]. Let X be a subspace of a separable Banach
space Y , J : X → Y be the natural embedding operator, and suppose X has the
Daugavet property. Then Y can be renormed so that the new norm coincides with
the original one on X and in the new norm ‖J + T‖ = 1 + ‖T‖ for every rank-1
operator T : X → Y .
The aim of our paper is to remove the separability condition in the above
theorem. On this way we introduce and study the following concept:
Definition 1.2. Let X and Y be Banach spaces. A linear continuous nonzero
operator G: X → Y is said to be a Daugavet center if the norm identity
‖G + T‖ = ‖G‖+ ‖T‖ (1.2)
is fulfilled for every rank-1 operator T : X → Y .
Our main result is more general than just the nonseparable version of Theorem
1.1. Namely, we prove the following:
Theorem 1.3. If G: X → Y is a Daugavet center, Y is a subspace of
a Banach space E, and J : Y → E is the natural embedding operator, then E
can be equivalently renormed in such a way that the new norm coincides with the
original one on Y , and J ◦G : X → E is also a Daugavet center.
Let us explain the structure of the paper. In Section 2. of this paper we collect
some straightforward generalizations to Daugavet centers of the properties known
for Id in the spaces with the Daugavet property. We also study the properties of
the unit ball images under Daugavet centers. In Section 3 we give some examples
of Daugavet centers quite different from the known before identity operator or
isometric embedding. Finally, Section 4. is devoted to the proof of the main
result.
In this paper we deal with real Banach spaces. We use the letters X, Y, E to
denote Banach spaces and their subspaces. L(X, Y ) stands for the space of all
linear bounded operators acting from X to Y . BX denotes the closed unit ball of
4 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Daugavet Centers
a Banach space X and SX denotes its unit sphere. For a bounded closed convex
set A ⊂ X and for x∗ ∈ X∗ we denote
S(A, x∗, ε) = {x ∈ A: x∗(x) ≥ supx∗(A)− ε}
the slice of A, generated by x∗. We use the notation
S(x∗, ε) = {x ∈ BX : x∗(x) ≥ 1− ε}
for the slice of BX determined by a functional x∗ ∈ SX∗ and ε > 0.
We say that an element x ∈ A is a denting point of the set A if for every
ε > 0 there is a slice of A which contains x and has a diameter smaller than ε.
A set A is said to have the Radon-Nikodým property if every closed convex
subset B ⊂ A is the closed convex hull of its denting points.
The operator T ∈ L(X,Y ) is said to be a strong Radon–Nikodým operator if
the closure of T (BX) has the Radon–Nikodým property.
2. Basic Properties of Daugavet Centers
Definition 1.2 implies the equality ‖aG+bT‖ = a‖G‖+b‖T‖ for every a, b ≥ 0.
This means that an operator G is a Daugavet center if and only if G/‖G‖ is.
Therefore below we mostly consider the case ‖G‖ = 1, and by the same reason,
when it is convenient, we require ‖T‖ = 1.
Theorem 2.1. For an operator G ∈ L(X, Y ) with ‖G‖ = 1 the following
assertions are equivalent:
(i) G is a Daugavet center.
(ii) For every y0 ∈ SY and every slice S(x∗0, ε0) of BX there is another slice
S(x∗1, ε1) ⊂ S(x∗0, ε0) such that for every x ∈ S(x∗1, ε1) the inequality
‖Gx + y0‖ > 2− ε0 holds.
(iii) For every y0 ∈ SY , x∗0 ∈ SX∗ and ε > 0 there is x ∈ BX such that
x∗0(x) ≥ 1− ε and ‖Gx + y0‖ > 2− ε.
(iv) For every x∗0 ∈ SX∗ and every weak∗ slice S(BY ∗ , y0, ε0) (where y0 ∈
SY ⊂ SY ∗∗) there is another weak∗ slice S(BY ∗ , y1, ε1) ⊂ S(BY ∗ , y0, ε0)
such that for every y∗ ∈ S(BY ∗ , y1, ε1) the inequality ‖G∗y∗+x∗0‖ > 2−ε0
holds.
(v) For every x∗0 ∈ SX∗ and every weak∗ slice S(BY ∗ , y0, ε0) there is y∗ ∈
S(BY ∗ , y0, ε0) which satisfies the inequality ‖G∗y∗ + x∗0‖ > 2− ε0.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 5
T. Bosenko and V. Kadets
P r o o f. (i) ⇒ (ii). Define T by Tx = x∗0(x)y0. Then ‖G∗ + T ∗‖ =
‖G+T‖ = 2, so there exists a functional y∗ ∈ SY ∗ such that ‖G∗y∗+T ∗y∗‖ ≥ 2−ε0
and y∗(y0) ≥ 0. Put
x∗1 =
G∗y∗ + T ∗y∗
‖G∗y∗ + T ∗y∗‖ , ε1 = 1− 2− ε0
‖G∗y∗ + T ∗y∗‖ .
Then for every x ∈ S(x∗1, ε1) we have
〈(G∗ + T ∗)y∗, x〉 ≥ (1− ε1)‖G∗y∗ + T ∗y∗‖ = 2− ε0;
hence
1 + x∗0(x) ≥ y∗(Gx) + y∗(y0)x∗0(x) ≥ 2− ε0,
which implies that x∗0(x) ≥ 1− ε0, i.e., x ∈ S(x∗0, ε0), and
2− ε0 ≤ y∗(Gx) + y∗(y0) = y∗(Gx + y0) ≤ ‖Gx + y0‖.
The implication (ii) ⇒ (iii) is evident. Let us prove (iii) ⇒ (i). If T is
a rank-1 operator and ‖T‖ = 1, then T can be represented as Tx = x∗0(x)y0 with
y0 ∈ SY , x∗0 ∈ SX∗ . Fix an ε > 0 and let x ∈ BX be the corresponding element
from (iii). Then
2− ε ≤ ‖Gx + y0‖ ≤ ‖Gx + x∗0(x)y0‖+ ‖(1− x∗0(x))y0‖
≤ ‖(G + T )x‖+ ε‖y0‖ ≤ ‖G + T‖+ ε.
So we have proved the equivalence (i) ⇔ (ii) ⇔ (iii). The remaining equi-
valence (i) ⇔ (iv) ⇔ (v) can be proved in the same way by using the adjoint
operators T ∗ and G∗ instead of T and G.
For a bounded subset A ⊂ Y denote ry(A) = sup{‖y − a‖: a ∈ A}.
Definition 2.2. A bounded subset A ⊂ Y is said to be a quasiball if for every
y ∈ Y
ry(A) = ‖y‖+ r0(A). (2.3)
Definition 2.3. A bounded subset A ⊂ Y is called antidentable if for every
y ∈ Y and for every r ∈ [0, ry(A))
conv (A \BY (y, r)) ⊃ A. (2.4)
Theorem 2.4. If an operator G ∈ SL(X,Y ) is a Daugavet center, then A :=
G(BX) is a quasiball.
6 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Daugavet Centers
P r o o f. From Theorem 2.1, item (iii) it follows in particular that for every
y ∈ Y and for every ε > 0 there is an x ∈ BX with ‖y −Gx‖ > ‖y‖+ 1− ε. So,
ry(A) > ‖y‖+ 1− ε ≥ ‖y‖+ r0(A)− ε.
Theorem 2.5. If an operator G ∈ SL(X,Y ) is a Daugavet center, then for
every y ∈ Y and for every r ∈ [0, ry(G(BX)))
V := conv
(
BX \G−1(BY (y, r))
) ⊃ BX . (2.5)
P r o o f. Assume it is not true. Then there is an y ∈ Y and an r ∈
[0, ry(G(BX)) such that the corresponding V does not contain the whole BX .
Consider a slice S = S(x∗, ε0) of BX which does not intersect V . For this slice
we have S ⊂ G−1(BY (y, r)). Select such a small δ > 0 that ‖y‖ + 1 − δ > r.
By Theorem 2.1, item (iii) applied to x∗0 = x∗, y0 = −y and ε = min{ε, δ}, there
is an x ∈ S ⊂ G−1(BY (y, r)) with ‖Gx−y‖ > ‖y‖+1−δ > r. But in such a case
Gx /∈ BY (y, r), i.e., x /∈ G−1(BY (y, r)), which leads to contradiction.
Corollary 2.6. If an operator G ∈ SL(X,Y ) is a Daugavet center, then
A := G(BX) is antidentable.
P r o o f. According to the previous theorem for every y ∈ Y and for every
r ∈ [0, ry(A)) the inclusion (2.5) holds true. So
A ⊂ G(V ) ⊂ conv G
(
BX \G−1(BY (y, r))
)
= conv (A \BY (y, r)) .
Now we prove that the properties from Theorems 2.4 and 2.5 together give
a characterization of Daugavet centers. In fact, we prove even more:
Theorem 2.7. An operator G ∈ SL(X,Y ) is a Daugavet center if and only if
it satisfies the following two conditions:
1. the set A := G(BX) is a quasiball;
2. the condition (2.5) holds true for all y ∈ Y and for all r ∈ [0, ry(G(BX))).
Moreover, if G is a Daugavet center, then equation (1.2) holds true for every
strong Radon–Nikodým operator T ∈ L(X,Y ).
P r o o f. What remains to prove is that conditions (1) and (2) imply equation
(1.2) for every strong Radon–Nikodým operator T ∈ L(X, Y ). Fix an ε > 0. Let
x ∈ SX be an element for which ‖Tx‖ > ‖T‖ − ε, and Tx belongs to a slice S̃ of
T (BX) with the diameter smaller than ε. Put r = rTx(G(BX))− ε. Then T−1S̃
is a slice of BX , so
T−1S̃
⋂(
BX \G−1(BY (Tx, r))
) 6= ∅.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 7
T. Bosenko and V. Kadets
Hence there is an x0 ∈ BX such that Tx0 ∈ S̃ (and, consequently,
‖Tx0 − Tx‖ < ε), but Tx0 /∈ BY (Tx, r), i.e, ‖Gx0 − Tx‖ > r. Then
‖G− T‖ ≥ ‖Gx0 − Tx0‖ ≥ ‖Gx0 − Tx‖ − ε > r − ε = rTx(G(BX))− 2ε
= ‖Tx‖+ r0(G(BX))− 2ε ≥ ‖Tx‖+ ‖G‖ − 2ε ≥ ‖T‖+ ‖G‖ − 3ε.
R e m a r k 2.8. Theorem 2.7 will not hold true if we require G(BX) to be
antidentable instead of condition (2) of this theorem. Consider G: C[0, 1]⊕1R→
C[0, 1], G(f, a) = f . It is obvious that G(BC[0,1]⊕1R) = BC[0,1]. Since C[0, 1] has
the Daugavet property, BC[0,1] is an antidentable quasiball. Let us show that G
is not a Daugavet center. Consider a rank -1 operator T : C[0, 1]⊕1 R→ C[0, 1],
T (f, a) = a · y0 for some y0 ∈ SC[0,1]. So ‖G‖+ ‖T‖ = 2, but
‖G + T‖ = sup
(f,a)∈SX
‖(G + T )(f, a)‖
= sup
(f,a)∈SX
‖f + a y0‖ ≤ sup
(f,a)∈SX
(‖f‖+ |a|) = 1.
For a set Γ denote by FIN(Γ) the set of all finite subsets of Γ. Recall that
a (maybe uncountable) series
∑
n∈Γ xn in a Banach space X is said to be uncon-
ditionally convergent to x ∈ X if for every ε > 0 there is an A ∈ FIN(Γ) such
that for every B ∈ FIN(Γ), B ⊃ A
‖x−
∑
n∈B
xn‖ < ε.
Theorem 2.9. Let G ∈ L(X, Y ). Suppose that inequality ‖G+T‖ ≥ C +‖T‖
with C > 0 holds for every operator T from a subspace M⊂ L(X, Y ) of operators.
Let T̃ =
∑
n∈Γ Tn be a (maybe uncountable) pointwise unconditionally convergent
series of operators Tn ∈M. Then ‖G− T̃‖ ≥ C.
P r o o f. Pointwise unconditional convergence of
∑
n∈Γ Tn implies that for
every x ∈ X
sup
{∥∥∥∥
∑
n∈A
Tnx
∥∥∥∥: A ∈ FIN(Γ)
}
< ∞.
Consequently, by the Banach-Steinhaus theorem, the quantity
α = sup
{∥∥∥∥
∑
n∈A
Tn
∥∥∥∥: A ∈ FIN(Γ)
}
8 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Daugavet Centers
is finite, and whenever B ⊂ Γ, then
∥∥∥∥
∑
n∈B
Tn
∥∥∥∥ ≤ sup
{∥∥∥∥
∑
n∈A
Tn
∥∥∥∥: A ∈ FIN(Γ), A ⊂ B
}
≤ α.
Let ε > 0 and pick A0 ∈ FIN(Γ) such that ‖∑
n∈A0
Tn‖ ≥ α− ε. Then we have
‖G− T̃‖ ≥
∥∥∥∥G−
∑
n∈A0
Tn
∥∥∥∥−
∥∥∥∥
∑
n/∈A0
Tn
∥∥∥∥ ≥ C +
∥∥∥∥
∑
n∈A0
Tn
∥∥∥∥− α ≥ C − ε,
which proves the theorem.
R e m a r k 2.10. Let G: X → Y be a Daugavet center. Since by Theorem 2.7
every weakly compact operator satisfies (1.2), the above theorem means, in par-
ticular, that G cannot be represented as a pointwise unconditionally convergent
series of weakly compact operators. So, neither X nor Y can have an uncondi-
tional basis (countable or uncountable) or be represented as unconditional sum of
reflexive subspaces.
Lemma 2.11. Let G: X → Y be a Daugavet center, ‖G‖ = 1. Then for every
finite-dimensional subspace Y0 of Y , every ε0 > 0 and every slice S(x∗0, ε0) of BX
there is a slice S(x∗1, ε1) ⊂ S(x∗0, ε0) of BX such that
‖y + tGx‖ ≥ (1− ε0)(‖y‖+ |t|) ∀y ∈ Y0, x ∈ S(x∗1, ε1), ∀t ∈ R. (2.6)
P r o o f. Let δ = ε0/2 and pick a finite δ-net {y1, . . . , yn} in SY0 .
By a repeated application of Theorem 2.1, item (ii), we obtain a sequence of
slices S(x∗0, ε0) ⊃ S(u∗1, δ1) ⊃ . . . ⊃ S(u∗n, δn) such that one has
‖yk + Gx‖ ≥ 2− δ (2.7)
for all x ∈ S(u∗k, δk). Put x∗1 = u∗n and ε1 = δn; then (2.7) is valid for every
x ∈ S(x∗1, ε1) and k = 1, . . . , n. This implies that for every x ∈ S(x∗1, ε1) and
every y ∈ SY0 the condition
‖y + Gx‖ ≥ 2− 2δ = 2− ε0
holds.
Let 0 ≤ t1, t2 ≤ 1 with t1 + t2 = 1. If t1 ≥ t2, we have for x and y as above
‖t1Gx + t2y‖ = ‖t1(Gx + y) + (t2 − t1)y‖ ≥ t1‖Gx + y‖ − |t2 − t1| ‖y‖
≥ t1(2− ε0) + t2 − t1 = t1 + t2 − t1ε0 ≥ 1− ε0,
and an analogous argument shows this estimate in the case t1 < t2.
This implies (2.6) by the homogeneity of the norm and the symmetry of SY0 .
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 9
T. Bosenko and V. Kadets
Theorem 2.12. Let G: X → Y be a Daugavet center. Then G fixes a copy
of `1.
P r o o f. Using Lemma 2.11 inductively, we construct the sequences of the
vectors {xn}∞n=1 ⊂ X and {yn}∞n=1 ⊂ Y and a sequence of the slices S(x∗n, εn),
εn ≤ 2−n, n ∈ N, such that yn = Gxn, xn ∈ S(x∗n, εn) and for every y ∈
lin{y1, . . . , yn} and every t ∈ R the inequality
‖y + tyn+1‖ ≥ (1− εn)(‖y‖+ |t|‖yn+1‖)
holds true. Hence the sequences {xn}∞n=1 ⊂ X and {yn}∞n=1 ⊂ Y are equivalent
to the canonical basis in `1, and G fixes a copy of `1.
3. Some Examples of Daugavet Centers
Proposition 3.1. Let G: X → Y be a Daugavet center. Then for all surjective
linear isometries V : X → X and U : Y → Y the operator UGV is also a Daugavet
center.
P r o o f. Let T : X → Y have rank one. Then
‖UGV + T‖ = ‖U(GV + U−1T )‖ = ‖U‖‖GV + U−1T‖
= ‖GV + U−1T‖ = ‖(G + U−1TV −1)V ‖
= ‖G + U−1TV −1‖‖V ‖ = ‖G + U−1TV −1‖.
The operator T can be represented as Tx = x∗0(x)y0 with y0 ∈ Y , x∗0 ∈ X∗ hence
U−1TV −1x = x∗0(V
−1x)U−1y0 is also a rank-1 operator. Since G is a Daugavet
center,
‖UGV + T‖ = ‖G + U−1TV −1‖ = ‖G‖+ ‖U−1TV −1‖
= ‖G‖+ ‖T‖ = ‖UGV ‖+ ‖T‖.
Proposition 3.2. Let G: X → Y be a Daugavet center. Then G̃ : X/KerG →
Y (the natural injectivization of G) is also a Daugavet center.
P r o o f. We will prove this proposition using Definition 1.2 of the Daugavet
center. Let T ∈ L(X/KerG,Y ) be a rank-1 operator and q: X → X/KerG be
the corresponding quotient mapping. Then the composition T ◦ q: X → Y is
a linear continuous rank-1 operator. Since G is a Daugavet center, the identity
‖G + T ◦ q‖ = ‖G‖+ ‖T ◦ q‖
10 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Daugavet Centers
holds true. The operator G̃ is the natural injectivization of G hence ‖G‖ = ‖G̃‖.
It is well known that q(
◦
BX) =
◦
BX/ Ker G, where
◦
BX and
◦
BX/ Ker G are the open
unit balls of X and X/KerG, respectively. This implies that ‖T ◦ q‖ = ‖T‖ and
‖G̃ + T‖ = ‖(G̃ + T ) ◦ q‖ = ‖G + T ◦ q‖. So we have
‖G̃ + T‖ = ‖G + T ◦ q‖ = ‖G‖+ ‖T ◦ q‖ = ‖G̃‖+ ‖T‖,
which proves the proposition.
Lemma 3.3. If G1: X1 → Y1 and G2: X2 → Y2 are Daugavet centers, ‖G1‖ =
‖G2‖ = 1. Then the operator G: X1⊕∞X2 → Y1⊕∞Y2 (G: X1⊕1X2 → Y1⊕1Y2),
which maps every (x1, x2) into (G1x1, G2x2), is a Daugavet center.
P r o o f. We first prove that G: X1⊕∞X2 → Y1⊕∞ Y2 is a Daugavet center.
Consider x∗j ∈ X∗
j , yj ∈ Yj (j = 1, 2) with ‖(y1, y2)‖ = max{‖y1‖, ‖y2‖} = 1,
‖(x∗1, x∗2)‖ = ‖x∗1‖ + ‖x∗2‖ = 1. Assume without loss of generality that
‖y1‖ = 1. We will use the characterization of the Daugavet centers from item
(iii) of Theorem 2.1. For a given ε > 0 there is an x1 ∈ X1 satisfying
‖x1‖ = 1, x∗1(x1) ≥ ‖x∗1‖(1− ε), ‖G1x1 + y1‖ ≥ 2− ε.
Also, pick x2 ∈ X2 such that
‖x2‖ = 1, x∗2(x2) ≥ ‖x∗2‖(1− ε).
Then ‖(x1, x2)‖ = 1, 〈(x∗1, x∗2), (x1, x2)〉 ≥ 1− ε and
‖G(x1, x2) + (y1, y2)‖ ≥ ‖Gx1 + y1‖ ≥ 2− ε.
Thus, G is a Daugavet center.
A similar calculation, based on item (v) of Theorem 2.1, proves that G: X1⊕1
X2 → Y1 ⊕1 Y2 is a Daugavet center.
Lemma 3.4. Let G: X → Y be a Daugavet center, ‖G‖ = 1. Denote G̃:
X → Y ⊕1 Y1, G̃x = (Gx, 0), and Ĝ: X1 ⊕∞ X → Y , Ĝ(x1, x2) = Gx2. Then:
(a) the operator G̃ is a Daugavet center;
(b) the operator Ĝ is a Daugavet center.
P r o o f. Part (b) can be proved in a similar manner as Lemma 3.3, so
we present only the proof of (a). Consider x∗ ∈ SX∗ , yj ∈ Yj (j = 0, 1) with
‖(y0, y1)‖ = ‖y0‖ + ‖y1‖ = 1. By Theorem 2.1 there is, given ε > 0, some
x0 ∈ S(x∗, ε) satisfying ∥∥∥∥Gx0 +
y0
‖y0‖
∥∥∥∥ ≥ 2− ε.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 11
T. Bosenko and V. Kadets
Then we have
‖G̃x0 + (y0, y1)‖ = ‖Gx0 + y0‖+ ‖y1‖
≥
∥∥∥∥Gx0 +
y0
‖y0‖ + y0
(
1− 1
‖y0‖
)∥∥∥∥ + ‖y1‖
≥
∥∥∥∥Gx0 +
y0
‖y0‖
∥∥∥∥ + ‖y0‖
(
1− 1
‖y0‖
)
+ ‖y1‖
≥ 2− ε
which proves the lemma.
Let K be a compact space without isolated points. Then C(K) has the
Daugavet property and this means that the identity operator is a Daugavet center.
Therefore, by Proposition 3.1, every surjective linear isometry V : C(K) → C(K)
is a Daugavet center.
In particular, if we consider any bijective continuous function ϕ: K → K,
then the operator Gϕ: C(K) → C(K), Gϕf = f ◦ϕ is a surjective linear isometry
and hence a Daugavet center.
Our next aim is to prove that for every continuous function ϕ: K → K such
that ϕ−1(t) is nowhere dense in K for all t ∈ K the corresponding operator Gϕ
is a Daugavet center as well.
Lemma 3.5. For an operator G: X → C(K), ‖G‖ = 1, the following asser-
tions are equivalent:
(i) G is a Daugavet center.
(ii) For every ε > 0, every open set U ⊂ K, every x∗ ∈ SX∗ and s = ±1
there is f ∈ S(x∗, ε) such that
sup
t∈U
s · (Gf)(t) > 1− ε.
P r o o f. (i) ⇒ (ii) Let us consider a function g ∈ SC(K) such that supp g ⊂ U
and s · g ≥ 0. By Theorem 2.1, for every ε > 0 and every x∗ ∈ SX∗ there is
an element f ∈ S(x∗, ε) such that
sup
t∈K
|(Gf + g)(t)| > 2− ε.
Notice that |Gf + g| = |Gf | ≤ 1 on K \ U and hence |Gf + g| attains
its supremum in U . Then there is a point t0 ∈ U which fulfills the inequality
|(Gf + g)(t0)| > 2− ε. Since s · g ≥ 0, then s · (Gf)(t0) ≥ 0 and
|(Gf + g)(t0)| = s · (Gf + g)(t0) ≤ sup
t∈U
s · (Gf + g)(t)
≤ sup
t∈U
s · (Gf)(t) + sup
t∈U
s · g(t) ≤ sup
t∈U
s · (Gf)(t) + 1.
12 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Daugavet Centers
Therefore
sup
t∈U
s · (Gf)(t) > 1− ε.
(ii) ⇒ (i) Consider g ∈ SC(K), pick some τ ∈ K with |g(τ)| = 1 and put s = g(τ).
Then for every ε > 0 there is an open neighborhood U of τ such that s · g > 1− ε
on U . For every x∗ ∈ SX∗ there is an element f ∈ S(x∗, ε) which satisfies the
inequality
sup
t∈U
s · (Gf)(t) > 1− ε.
Then we have
‖Gf + g‖ = sup
t∈K
|s · (Gf + g)(t)| ≥ sup
t∈U
(s · (Gf)(t) + s · g(t))
≥ sup
t∈U
s · (Gf)(t) + 1− ε ≥ 1− ε + 1− ε = 2− 2ε.
By Theorem 2.1, G is a Daugavet center.
In Lemma 3.5 consider X = C(K1). By the Riesz representation theorem, for
any linear functional x∗ on C(K1) there is a unique Borel regular signed measure
σ on K1 such that
x∗(f) =
∫
K1
f dσ
for all f ∈ C(K1), and ‖x∗‖ = |σ|(K1). So every slice
S(x∗, ε) = {f ∈ BC(K1):
∫
K1
f dσ ≥ |σ|(K1)− ε}
= {f ∈ BC(K1):
∫
K1
(1− f (1K+
1
− 1K−
1
)) d|σ| ≤ ε}.
Here K1 = K+
1 t K−
1 is a Hahn decomposition of K1 for σ, and 1A denotes
a characteristic function of the set A.
So, in the case of X = C(K1) Lemma 3.5 can be reformulated as follows:
Lemma 3.6. For an operator G: C(K1) → C(K2), ‖G‖ = 1, the following
assertions are equivalent:
(i) G is a Daugavet center.
(ii) For every ε > 0, every open set U ⊂ K2 and every Borel regular signed
measure σ on K1 and s = ±1 there is an f ∈ BC(K1) such that
∫
K1
(1− f (1K+
1
− 1K−
1
)) d|σ| ≤ ε (3.8)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 13
T. Bosenko and V. Kadets
and
sup
t∈U
s · (Gf)(t) > 1− ε. (3.9)
Theorem 3.7. Let K1 and K2 be compact spaces without isolated points, ϕ:
K2 → K1 be a continuous function such that for every t ∈ K1 the set ϕ−1(t) is
nowhere dense in K2. Suppose that an operator Gϕ: C(K1) → C(K2) maps every
f ∈ C(K1) into the composition f ◦ ϕ. Then Gϕ is a Daugavet center.
P r o o f. Consider an ε > 0, an open set U ⊂ K2, and a Borel regular
signed measure σ on K1, and put s = 1. We will construct a function f ∈ BC(K1)
satisfying (3.8) and (3.9).
The measure σ can have at most countable set of atoms. Let us show that
for every open U ⊂ K2 the set ϕ(U) is uncountable. Assume that there exists
an open set U ⊂ K2 for which it is not true. Then ϕ−1(ϕ(U)) is a countable
union of nowhere dense sets in K2 because for every t ∈ ϕ(U) ⊂ K1 the set
ϕ−1(t) is nowhere dense in K2 by the condition of this theorem. This contradicts
the Baire category theorem.
So, we can pick a point t0 ∈ U such that ϕ(t0) is not an atom of σ, i.e.,
|σ|(ϕ(t0)) = 0. Moreover, since σ is a Borel regular measure, there is an open
neighborhood V ⊂ ϕ(U) of the point ϕ(t0) such that |σ|(V ) < ε/4.
Now we pass on to the construction of f . To satisfy (3.9) we select f in
such a way that f(ϕ(t0)) > 1 − ε. First, we pick a function f̃ ∈ S(σ, ε/2).
If f̃(ϕ(t0)) > 1− ε, then we can simply put f = f̃ .
If f̃(ϕ(t0)) ≤ 1 − ε, we put f = f̃ in K1 \ V and f(ϕ(t0)) = 1. Since
K1 \ V ∪ {ϕ(t0)} is closed, we can use the Tietze extension theorem to construct
a continuous extension f on V \ ϕ(t0) and keep the condition ‖f‖ = 1. Now we
show that (3.8) also holds for this f :
∫
K1
(1− f (1K+
1
− 1K−
1
)) d|σ| =
∫
K1\V
(1− f̃ (1K+
1
− 1K−
1
)) d|σ|
+
∫
V
(1− f (1K+
1
− 1K−
1
)) d|σ|
≤
∫
K1
(1− f̃ (1K+
1
− 1K−
1
)) d|σ|+ ε/2
≤ ε/2 + ε/2 = ε.
So, for every ε > 0, every Borel regular measure σ on K1, every open set
U ⊂ K2 and s = 1 we have a function f ∈ BC(K1) satisfying inequalities (3.8)
14 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Daugavet Centers
and (3.9). The case s = −1 can be proved in a very similar way. Thus, by Lemma
3.6, Gϕ is a Daugavet center.
Let us give an example of a Daugavet center on C(K) of a very different
nature.
Proposition 3.8. Consider K1 = [−1; 1] and define G: C(K1) → C(K1) as
(Gf)(x) = f(x)+f(−x)
2 . Then G is a Daugavet center.
P r o o f. We will use Lemma 3.6 to prove this proposition. Let us fix
an ε > 0, a Borel regular signed measure σ, an open set U ⊂ K1, s = 1 and
a function f̃ ∈ S(σ, ε/2). If there is a point t0 ∈ U such that f̃(t0)+f̃(−t0)
2 > 1− ε,
then
sup
t∈U
s · (Gf̃)(t) > 1− ε.
Otherwise we pick a point t1 ∈ U such that neither t1 nor −t1 is an atom of σ.
Consider disjoint segments [a1, b1], [a2, b2] ⊂ K1 such that |σ|([a1, b1]) < ε/8,
t1 ∈ [a1, b1] and |σ|([a2, b2]) < ε/8, −t1 ∈ [a2, b2]. Let f̃1: [a1, b1] → K1 be
a continuous function such that f̃1(a1) = f̃(a1), f̃1(b1) = f̃(b1) and f̃1(t1) = 1.
Let f̃2: [a2, b2] → K1 be a continuous function such that f̃2(a2) = f̃(a2), f̃2(b2) =
f̃(b2) and f̃2(−t1) = 1. Then denote ∆ := K1 \ {[a1, b1] ∪ [a2, b2]}, put
f := 1[a1,b1]f̃1 + 1[a2,b2]f̃2 + 1∆f̃ .
Then f ∈ BC(K1) and
∫
K1
(1− f (1K+
1
− 1K−
1
)) d|σ| =
∫
∆
(1− f̃ (1K+
1
− 1K−
1
)) d|σ|
+
∫
[a1,b1]
(1− f̃1 (1K+
1
− 1K−
1
)) d|σ| +
∫
[a2,b2]
(1− f̃2 (1K+
1
− 1K−
1
)) d|σ|
≤
∫
K1
(1− f̃ (1K+
1
− 1K−
1
)) d|σ| + ε/4 + ε/4 ≤ ε/2 + ε/2 = ε.
Hence f ∈ S(σ, ε) and
sup
t∈U
s · (Gf)(t) ≥ f(t1) + f(−t1)
2
= 1.
If we put s = −1, the analogous conclusions prove the proposition.
A rather nontrivial class of the Daugavet centers was discovered in [9], where
every isometric embedding G: L1[0, 1] → L1[0, 1] was a Daugavet center. Let us
show that the analogous result for C[0, 1] is false. This will answer in negative
a question from [10].
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 15
T. Bosenko and V. Kadets
E x a m p l e. Consider T : C[0, 1] → C[0, 1],
Tf =
{
f(2t) if t ∈ [
0, 1
2
]
,
2f(1)(1− t) if t ∈ (
1
2 , 1
]
.
Let us prove that T is an isometric embedding. It is obvious that T is a linear
operator. Notice that |Tf | attains its supremum in
[
0, 1
2
]
because for
every t ∈ (
1
2 , 1
]
we have |Tf(t)| = |2f(1)(1− t)| < |f(1)| = |Tf
(
1
2
) |. Hence for
every f ∈ C[0, 1]
‖Tf‖ = sup
t∈[0, 1
2 ]
|Tf(t)| = sup
t∈[0,1]
|f(t)| = ‖f‖.
Now we show with the help of Lemma 3.5 that T is not a Daugavet center.
Our aim is to find an ε > 0, an open set U ⊂ [0, 1] and an x∗ ∈ SC∗[0,1] such
that every f ∈ S(x∗, ε) satisfies supt∈U Tf(t) ≤ 1 − ε. If we put ε := 1
4 and
U :=
(
3
4 , 1
]
, then for every f ∈ BC[0,1] we have
sup
t∈U
Tf(t) = sup
t∈( 3
4
,1]
2f(1)(1− t) ≤ 2|f(1)|
(
1− 3
4
)
=
|f(1)|
2
≤ 1
2
< 1− ε.
4. The Main Result
Definition 4.1. Let E be a seminormed space, A ⊂ BE, U be a free ultrafilter
on a set Γ, and f : Γ → A be a function. The triple (Γ,U , f) is said to be
an A-valued E-atom if for every w ∈ E
lim
U
‖f + w‖ = 1 + ‖w‖. (4.1)
The following characterization of Daugavet centers is a consequence of Theo-
rem 2.1 and Lemma 2.11.
Theorem 4.2. Let X, Y be Banach spaces. An operator G ∈ SL(X,Y ) is
a Daugavet center if and only if for every slice S of BX there is a G(S)-valued
Y-atom.
P r o o f. Let us start with the “if” part. We are going to prove that G
satisfies condition (iii) of Theorem 2.1. Fix y0 ∈ SY , x∗0 ∈ SX∗ and ε > 0. Denote
S = S(x∗0, ε). Due to our assumption there is a G(S)-valued Y-atom (Γ,U , f).
Plugging w = y0 in (4.1), we get, in particular, that ‖f(t) + y0‖ > 2− ε for some
t ∈ Γ. Since f(t) ∈ G(S), there is an x ∈ S such that f(t) = Gx. This x fulfills
the required conditions x∗0(x) ≥ 1 − ε and ‖Gx + y0‖ > 2 − ε. The “if” part is
proved.
16 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Daugavet Centers
Let us demonstrate the “only if” part. Fix a slice S of BX . Put Γ = FIN(Y )
and take the natural filter F on Γ whose base is formed by the collection of subsets
 ⊂ FIN(Y ), A ∈ FIN(Y ), where  := {B ∈ FIN(Y ): A ⊂ B}. According to
Lemma 2.11 for every A ∈ FIN(Y ) there is an element x(A) ∈ S such that for all
y ∈ A
‖y + G(x(A))‖ >
(
1− 1
|A|
)
(‖y‖+ 1).
Define f(A) := G(x(A)). It is easy to see that for every ultrafilter U Â F the
triple (Γ,U , f) is the required G(S)-valued Y -atom.
It is clear that if A ⊂ B, then every A-valued E-atom is at the same time
a B-valued E-atom. A BE-valued E-atom will be called just the E-atom.
Lemma 4.3. Let (E, p) be a seminormed space, Y be a subspace of E, and
(Γ,U , f) be a Y -atom. Define
pr(x) = U- lim
t
p(x + rf(t))− r
for x ∈ E and r > 0. Then:
(a) 0 ≤ pr(x) ≤ p(x) for all x ∈ E,
(b) pr(y) = p(y) for all y ∈ Y ,
(c) x 7→ pr(x) is convex for each r,
(d) r 7→ pr(x) is convex for each x,
(e) pr(tx) = tpr/t(x) for each t > 0.
P r o o f. The only thing that is not obvious is that pr ≥ 0; note that (b) is
just the definition of Y -atom. Now, given ε > 0, pick tε such that p(f(tε)) > 1−ε,
and
p(x + rf(tε)) ≤ U- lim
t
p(x + rf(t)) + ε.
Then
U- lim
t
p(x + rf(t)) ≥ U- lim
t
p(−rf(tε) + rf(t))− p(x + rf(tε))
= rp(f(tε)) + r − p(x + rf(tε))
≥ 2r − rε− U- lim
t
p(x + rf(t))− ε;
hence U- limt p(x + rf(t)) ≥ 1
2(2r − ε− rε) and pr(x) ≥ 0.
Lemma 4.4. Assume the conditions of Lemma 4.3. Then r 7→ pr(x) is
decreasing for each x. The quantity
p̄(x) := lim
r→∞ pr(x) = inf
r
pr(x)
satisfies (a)–(c) of Lemma 4.3 and, moreover,
p̄(tx) = tp̄(x) for t > 0, x ∈ X. (4.2)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 17
T. Bosenko and V. Kadets
P r o o f. By Lemma 4.3(a) and (d), r 7→ pr(x) is bounded and convex, hence
decreasing. Therefore, p̄ is well defined. Clearly, (4.2) follows from (e) above.
P r o o f of the main theorem (Theorem 1.3). Let G: X → Y be a Daugavet
center, Y be a subspace of a Banach space E, and J : Y → E be the natural
embedding operator.
Let P be the family of all seminorms q on E that are dominated by the norm
of E and for which q(y) = ‖y‖ for y ∈ Y . By Zorn’s lemma, P contains a minimal
element, say, p.
Claim. Every Y -atom (Γ,U , f) is at the same time an (E, p)-atom, i.e., for every
w ∈ E
lim
U
p(f + w) = 1 + p(w). (4.3)
P r o o f. To prove the claim, associate the functional p̄ from Lemma 4.4 to
p and (Γ,U , f). Notice that 0 ≤ p̄ ≤ p, but p̄ need not be a seminorm. However,
q(x) =
p̄(x) + p̄(−x)
2
defines a seminorm, and q ≤ p. By Lemma 4.3(b) and by minimality of p, we get
that
q(x) = p(x) ∀x ∈ X. (4.4)
Now, since p(x) ≥ p̄(x) and p(x) = p(−x) ≥ p̄(−x), (4.4) implies that p(x) =
p̄(x). Finally, by Lemma 4.3(a) and the definition of p̄, we have p(x) = pr(x) for
all r > 0; in particular p(x) = p1(x), which is our claim (4.3).
Now let us introduce a new norm on E as
|||x||| := p(x) + ‖[x]‖E/Y ;
and let us show that this is the equivalent norm that we need. Indeed, clearly
|||x||| ≤ 2‖x‖. On the other hand, |||x||| ≥ 1
3‖x‖. To see this assume ‖x‖ = 1.
If ‖[x]‖E/Y ≥ 1
3 , there is nothing to prove. If not, pick y ∈ Y such that
‖x− y‖ < 1
3 . Then p(y) = ‖y‖ > 2
3 , and
|||x||| ≥ p(x) ≥ p(y)− p(x− y) >
2
3
− ‖x− y‖ >
1
3
.
Therefore, ‖ · ‖ and ||| · ||| are equivalent norms. Also evidently for y ∈ Y
|||y||| = p(y) = ‖y‖.
What remains to prove is that J ◦ G : X → E is a Daugavet center. This
can be done easily with the help of Theorem 4.2 and Claim. Namely, let S be
18 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
Daugavet Centers
an arbitrary slice of BX . Due to Theorem 4.2 it is sufficient to demonstrate the
existence of a G(S)-valued (E, ||| · |||)-atom. Since G : X → Y is a Daugavet
center, the same Theorem 4.2 ensures the existence of a G(S)-valued Y -atom
(Γ,U , f). But according to Claim, (Γ,U , f) is also an (E, p)-atom. Consequently,
for every w ∈ E
lim
U
|||f + w||| = lim
U
(
p(f + w) + ‖[f + w]‖E/Y
)
= lim
U
p(f + w) + ‖[w]‖E/Y = 1 + p(w) + ‖[w]‖E/Y = 1 + |||w|||.
This means that (Γ,U , f) is the required G(S)-valued (E, ||| · |||)-atom. The main
theorem is proved. The same renorming idea is applicable to the theory
of `1-types [5].
The next corollary improves the statement of Remark 2.10.
Corollary 4.5. If G: X → Y is a nonzero Daugavet center, then neither X
nor Y can be embedded into a space E, in which the identity operator IdE has
a representation as a pointwise unconditionally convergent series of weakly com-
pact operators. In particular, neither X nor Y can be embedded into a space E
having an unconditional basis (countable or uncountable) or having a representa-
tion as unconditional sum of reflexive subspaces.
P r o o f. Let IdE =
∑
n∈Γ Tn, where the series is pointwise unconditionally
convergent, and all the Tn: E → E are weakly compact. At first, assume Y ⊂
E, and denote J ∈ L(Y, E) the natural embedding operator. Equip E with
the equivalent norm from Theorem 1.3 making J ◦ G a Daugavet center. Then
J ◦G =
∑
n∈Γ Tn ◦ J ◦G, the series is pointwise unconditionally convergent, and
all the operators Tn ◦ J ◦G are weakly compact. This contradicts Theorem 2.9.
Now assume X ⊂ E. Recall that for a set ∆ of big cardinality (say, for
∆ = BY ∗), there is an isometric embedding J : Y → `∞(∆). Since `∞(∆) is
an injective space (i.e., the Hahn–Banach extension theorem holds true
for `∞(∆)-valued operators), there is an operator U : E → `∞(∆) such that
U |X = J ◦G. Then
J ◦G = (U ◦ IdE)|X =
∑
n∈Γ
U ◦ (Tn)|X .
This representation leads to contradiction in the same way as in the previous
case.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 19
T. Bosenko and V. Kadets
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20 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
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| id | nasplib_isofts_kiev_ua-123456789-106629 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T15:52:42Z |
| publishDate | 2010 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Bosenko, T. Kadets, V. 2016-10-01T15:04:24Z 2016-10-01T15:04:24Z 2010 Daugavet Centers / T. Bosenko, V. Kadets // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 1. — С. 3-20. — Бібліогр.: 14 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106629 An operator G: X → Y is said to be a Daugavet center if ||G + T|| = ||G|| + ||T|| for every rank-1 operator T: X → Y . The main result of the paper is: if G: X →! Y is a Daugavet center, Y is a subspace of a Banach space E, and J : Y → E is the natural embedding operator, then E can be equivalently renormed in such a way that J ○ G : X → E is also a Daugavet center. This result was previously known for the particular case X = Y, G = Id and only in separable spaces. The proof of our generalization is based on an idea completely di®erent from the original one. We also give some geometric characterizations of the Daugavet centers, present a number of examples, and generalize (mostly in straightforward manner) to Daugavet centers some results known previously for spaces with the Daugavet property. Research of the second named author was conducted during his stay in the University of Granada and was supported by Junta de Andalucia grant P06-FQM-01438. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Daugavet Centers Article published earlier |
| spellingShingle | Daugavet Centers Bosenko, T. Kadets, V. |
| title | Daugavet Centers |
| title_full | Daugavet Centers |
| title_fullStr | Daugavet Centers |
| title_full_unstemmed | Daugavet Centers |
| title_short | Daugavet Centers |
| title_sort | daugavet centers |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106629 |
| work_keys_str_mv | AT bosenkot daugavetcenters AT kadetsv daugavetcenters |