Acta Mathematica Sinica, English Series Dec., 2010, Vol. 26, No. 12, pp. 2259–2268 Published online: November 15, 2010 DOI: 10.1007/s10114-010-8638-x Http://www.ActaMath.com
Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2010
A Variational Approach to Norm Attainment of Some Operators and Polynomials Mar´ıa D. ACOSTA
Jer´ onimo ALAMINOS
Departamento de An´ alisis Matem´ atico, Universidad de Granada, 18071 Granada, Spain E-mail :
[email protected] [email protected]
Domingo GARC´ IA
Manuel MAESTRE
Departamento de An´ alisis Matem´ atico, Universidad de Valencia, 46100 Valencia, Spain E-mail :
[email protected] [email protected] Abstract If X is an Asplund space, then every uniformly continuous function on BX ∗ which is holomorphic on the open unit ball, can be perturbed by a w∗ continuous and homogeneous polynomial on X ∗ to obtain a norm attaining function on the dual unit ball. This is a consequence of a version of Bourgain–Stegall’s variational principle. We also show that the set of N -homogeneous polynomials between two Banach spaces X and Y whose transposes attain their norms is dense in the corresponding space of N -homogeneous polynomials. In the case when Y is the space of Radon measures on a compact K, this result can be strengthened. Keywords
Banach space, variational principle, operator, polynomial, holomorphic function
MR(2000) Subject Classification
1
46B99, 46G20, 46G25
Introduction
Bishop and Phelps [1] proved that every closed convex and bounded set A of a Banach space X satisfies that the subset of support functionals of A is dense in the dual space. Bourgain [2, 3] proved a strengthening of this result for Banach spaces with the Radon–Nikod´ ym property. A new proof of this result is due to Stegall (see [4] and [5]). These authors established that for every closed and bounded Radon–Nikod´ ym subset A of a Banach space X, and every function Φ : A → R upper semicontinuous and bounded above, the set of elements x∗ in X ∗ such that Φ + Re x∗ strongly exposes A is a Gδ -dense set in X ∗ . Therefore, the Stegall’s result can be applied to dual Radon–Nikod´ ym Banach spaces. In such a case, we perturb by an element of the bidual. Since X is an Asplund space if, and only if, X ∗ is w∗ -dentable then we have a chance to perturb by something better, an element of X. In the first section we prove that, in fact, the result of denseness also holds by using w∗ -continuous functionals on X ∗ . In this proof we use directly the definition of Asplund space in terms of Fr´echet differentiability and the well-known duality between Fr´echet differentiability and strongly exposed points. Received December 18, 2008, revised March 18, 2010, accepted June 13, 2010 The first and second authors are supported by MEC Project MTM-2009–07498 and Junta de Andaluc´ıa “Proyecto de Excelencia” FQM–4911; the third and fourth authors are partially supported by MEC and FEDER Project MTM2008-03211; the fourth author is also supported by Prometeo 2008/101
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As a direct application of this perturbation result, we obtain that we can approximate holomorphic functions on the open unit ball of the dual of a complex Asplund Banach space which are bounded and uniformly continuous functions on the closed unit ball by norm attaining ones. In the last section, we deal with the density of norm attaining polynomials. In this case we use a variational principle due to Poliquin and Zizler [6] to obtain the density of the norm attaining N -homogeneous polynomials in the space of all N -homogeneous polynomials. In the same vein, it is worth mentioning the previous results by Finet and Georgiev [7] and Fabian and Zizler [8] on perturbation of polynomials. For some other kind of results relating isomorphic properties of a Banach space X and properties of the space of polynomials on X or the study of variational problems on Banach spaces, see [9–11]. 2
A Perturbation Result
The proof of the perturbation result that we announced in the introduction will be split in several lemmas. The argument follows the one due to Stegall [5, Theorem 14], and in our case it is more direct than the original one. A more general result due to Fabian and Zizler can be seen in [8, Theorem 5] (see also [12, Theorem 10.20]). Lemma 2.1 Let X be a (real) Banach space, E ⊂ X ∗ a bounded and closed subset and Φ : E → R bounded above and upper-semicontinuous. If we define the subset D by t ∗ ∗ , −1) : t ∈ [0, 1], x ∈ E , D := (x 1 + M − Φ(x∗ ) where M = sup Φ(E), then D is (norm) closed. Proof Assume that (y ∗ , s) is an element in the closure of D, and so there is a sequence in D such that tn ∗ (x , −1) → (y ∗ , s), (2.1) 1 + M − Φ(x∗n ) n where tn ∈ [0, 1], x∗n ∈ E for every n. We can assume that the sequence {tn } converges to t. If t = 0, since E is bounded, then (y ∗ , s) = (0, 0) ∈ D. Otherwise t ∈]0, 1] and −t → s, 1 + M − Φ(x∗n ) t x∗ n hence the sequence {Φ(x∗n )} converges. By (2.1), since the sequence 1+Mn−Φ(x converges ∗) n ∗ ∗ ∗ to y , then the sequence {xn } converges to an element z ∈ E satisfying tz ∗ = y∗ . 1 + M − lim Φ(x∗n )
(2.2)
Since Φ is upper-semicontinuous, then there holds that Φ(z ∗ ) ≥ lim{Φ(x∗n )}. If we take α=
t (1 + M − Φ(z ∗ )), 1 + M − lim Φ(x∗n )
(2.3)
A Variational Approach to Norm Attainment
by (2.3), α ∈ [0, 1] and (y ∗ , s) =
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α (z ∗ , −1) 1 + M − Φ(z ∗ )
and so (y ∗ , s) ∈ D.
Let us recall that for a bounded and closed subset A of a Banach space X, a bounded above function g : A → R strongly exposes A if g attains its maximum at a point x0 ∈ A, and {xn } → x0 for every sequence {xn } ⊂ A such that {g(xn )} → max g(A). Lemma 2.2 by
Under the same assumptions of Lemma 2.1, we define the function ϕ : X → R ϕ(x) := sup{Φ(x∗ ) + x∗ (x) − M − 1 : x∗ ∈ E},
x ∈ X.
If an element (x, ϕ(x)) ∈ X × R strongly exposes D, then Φ + x strongly exposes E. Proof Assume that (x0 , ϕ(x0 )) strongly exposes D, so it attains its maximum at D. Since the value of this function at (0, 0) is zero, this maximum is nonnegative. If the maximum is ∗ t0 attained at the element 1+M −Φ(x ∗ ) x0 , −1 ∈ D, then it is satisfied that 0
t0 1 (x∗ (x0 ) − ϕ(x0 )) ≥ (x∗ (x0 ) − ϕ(x0 )), 1 + M − Φ(x∗0 ) 0 1 + M − Φ(x∗0 ) 0 and so we can assume that t0 = 1. We know that x∗ (x0 ) − ϕ(x0 ) x∗ (x0 ) − ϕ(x0 ) ≤ 0 ≤ 1, ∗ 1 + M − Φ(x ) 1 + M − Φ(x∗0 )
∀ x∗ ∈ E.
(2.4)
From the definition of ϕ, it follows that sup
x∗ ∈E
x∗ (x0 ) − ϕ(x0 ) = 1. 1 + M − Φ(x∗ )
(2.5)
From (2.4) and (2.5) it follows that the function of x∗ appearing in (2.4) attains its supremum at x∗0 ∈ E and so, we have Φ(x∗0 ) + x∗0 (x0 ) − 1 − M = ϕ(x0 ).
(2.6)
By the definition of ϕ, that means that the function defined on E by x∗ → Φ(x∗ )+x∗ (x)−M −1 attains its maximum at x∗0 , that is, Φ + x0 also attains its maximum on E (at x∗0 ). Assume that {x∗n } is a sequence in E such that Φ(x∗n )+x∗n (x0 ) converges to Φ(x∗0 )+x∗0 (x0 ). Then we have ∗ 1 − xn (x0 ) − ϕ(x0 ) ≤ |1 + M − Φ(x∗n ) − x∗n (x0 ) + ϕ(x0 )| (by (2.6)) ∗ 1 + M − Φ(x ) n
= |Φ(x∗0 ) + x∗0 (x0 ) − Φ(x∗n ) − x∗n (x0 )| → 0. 1 ∗ By using that (x0 , ϕ(x0 )) strongly exposes D at 1+M −Φ(x ∗ ) (x0 , −1), one has 0 (x∗0 , −1) (x∗n , −1) → . ∗ 1 + M − Φ(xn ) 1 + M − Φ(x∗0 )
As a consequence, {x∗n } → x∗0 , hence Φ + x0 strongly exposes E at x∗0 .
The next result goes back to Smulyan (see for instance [13, Proposition 5.11]) and we include it for completeness.
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Proposition 2.3 Let X be a real Banach space and D a bounded and closed subset of X ∗ . Suppose we define σD (x) = sup{x∗ (x) : x∗ ∈ D}. If σD is Fr´echet differentiable at a point x ∈ X, then x strongly exposes D. Proof By assumption, there is an element x∗0 ∈ X ∗ such that for every ε > 0, there exists δ > 0 satisfying y ∈ X, y ≤ δ ⇒ σD (x + y) − σD (x) − x∗0 (y) < ε y . We fix α = εδ and z ∈ SX . If x∗ ∈ D satisfies x∗ (x) > σD (x) − α, then we obtain 1 ∗ 1 (x − x∗0 )(δz) = [x∗ (x + δz) − x∗ (x) − x∗0 (δz)] δ δ 1 ≤ [σD (x + δz) − σD (x) + σD (x) − x∗ (x) − x∗0 (δz)] δ α 1 ≤ [ε δz + σD (x) − x∗ (x)] < ε + = 2ε. δ δ ∗ Since z is any element in the unit sphere, we obtain that x − x∗0 ≤ 2ε, and so x strongly exposes D at x∗0 . (x∗ − x∗0 )(z) =
Lemma 2.4 Let X be a Banach space, W a dense subset of X × R such that R+ W ⊂ W and ϕ : X → R is a convex and continuous function satisfying ϕ(0) < 0. Then the subset W ∩ Graf ϕ is dense in Graf ϕ. Proof Since ϕ is continuous, the subset {(y, t) ∈ X × R : ϕ(y) > t} is open. Let x0 be an element in X. By the assumption W is dense in X × R and so there is a sequence of elements {(xn , tn )} in W converging to (x0 , ϕ(x0 )) such that ϕ(xn ) > tn ,
∀ n ∈ N.
(2.7)
By using that ϕ is continuous, inequality (2.7) and ϕ(0) < 0, for every n, we can find a real number 0 < ρn < 1 satisfying ϕ(ρn xn ) = ρn tn ,
∀ n ∈ N.
(2.8)
By the assumption R+ W ⊂ W , so the elements ρn (xn , tn ) belongs to Graf ϕ ∩ W. We just need to prove that {ρn } → 1. Otherwise, we can assume that there is 0 ≤ ρ < 1 such that {ρn } → ρ. Then from (2.8) it follows that ϕ(ρx0 ) = ρϕ(x0 ), and this is a contradiction, since ϕ(0) < 0 and ϕ is convex and so we know that ϕ(ρx0 ) ≤ ρϕ(x0 ) + (1 − ρ)ϕ(0) < ρϕ(x0 ).
The next lemma is essentially well known, but we include it here for the sake of completeness. Lemma 2.5 Let X be a real Banach space, E ⊂ X ∗ a bounded and (norm) closed subset and Φ : E → R a bounded above function. Then the set {x ∈ X : Φ + x strongly exposes E}
A Variational Approach to Norm Attainment
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is a Gδ set. Proof
For x ∈ X and α > 0 we will write M (x) := sup{Φ(x∗ ) + x∗ (x) : x∗ ∈ E}
and S(x, α) := {x∗ ∈ E : Φ(x∗ ) + x∗ (x) > M (x) − α}. It is clear that the function Φ + x strongly exposes E if, and only if, limα→0 diam S(x, α) = 0. For every positive integer n, we define 1 . V (n) := x ∈ X : ∃ α > 0, diam S(x, α) < n We will prove that V (n) is open. Let us fix an element x ∈ V (n) and α > 0 with diam S(x, α) < 1 α α ∗ ∗ n . We choose K > sup{ x : x ∈ E}, β = 3 and y ∈ X satisfying that x − y < 3K and we will check the inclusion S(y, β) ⊂ S(x, α). (2.9) If x∗ ∈ S(y, β) , we obtain that Φ(x∗ ) + x∗ (x) = Φ(x∗ ) + x∗ (y) + x∗ (x) − x∗ (y) > M (y) − β − K x − y ≥ M (x) − β − 2K x − y α α ≥ M (x) − α. ≥ M (x) − − 2K 3 3K Because of the condition (2.9) we know that 1 , n and so y ∈ V (n). The subset {x ∈ X : Φ + x strongly exposes E} coincides with n∈N V (n) and the proof is finished. diam S(y, β)
0, there exists x0 ∈ X with x0 < ε such that h + p attains its norm on X ∗ , where p(x∗ ) = x∗ (x0 ), for all x∗ ∈ X ∗ . This corollary is stated as Corollary 3.2 in [15] in greater generality perturbing by N homogenous polynomials instead of linear forms, but we do not know if it holds true when the perturbation is in fact an N -homogenous polynomial. As a consequence, we have the following corollary that means a Bishop–Phelps type result. Corollary 2.8 Let X be an Asplund space. Then the set of norm attaining elements of Au (BX ∗ ) is dense in Au (BX ∗ ). Furthermore, w∗ -continuous holomorphic functions can be approximated by norm attaining functions. Corollary 2.9 Let X be an Asplund space. Then the w∗ -continuous functions of Au (BX ∗ ) can be approximated by norm attaining elements of Au (BX ∗ ) which are w∗ -continuous functions.. 3
Norm Attaining Polynomials
Now we will use some other perturbed optimization results in order to obtain consequences for polynomials whose transposes attain the norm. We will denote by P(N X, Y ) the space of all bounded N -homogeneous polynomials from X to Y , that is, the restrictions to the diagonal of N -linear and bounded mappings from X N into Y . The above space is a Banach space under the norm P = sup{ P x : x ∈ BX },
P ∈ P(N X, Y ).
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We recall that P ∈ P(N X, Y ) is of finite rank if there are k ∈ N, x∗i ∈ X ∗ , xi ∈ X (1 ≤ i ≤ k) such that k
P (x) = x∗i (x)N xi , ∀ x ∈ X. i=1
The closure of the space of finite rank N -homogeneous polynomials from X to Y is the space PA (N X, Y ) of approximable polynomials. If X ∗ has the approximation property, then PA (N X, Y ) coincides with Pw (N X, Y ) (see [16, Proposition 2.8]), for any N ∈ N, where Pw (N X, Y ) denotes the space of all N -homogeneous polynomials from X into Y which are weakly continuous on the unit ball of X. Given P ∈ P(k X, Y ), the transpose P t of P is the linear mapping P t : Y ∗ → P(k X) defined by P t (y ∗ )(x) = y ∗ (P (x)). For the linear case (k = 1), P t is the usual adjoint operator. Notice that P t = P and P t attains its norm whenever P does it. There is a variational principle along the same line proved in Theorem 2.6 for functions defined on a w∗ -compact set which are convex, Lipschitz and w∗ -lower semicontinuous due to Poliquin and Zizler. Theorem 3.1 ([6, Theorem 1]) Let ϕ be a w∗ -lower semicontinuous convex and Lipschitz function defined on a w∗ -compact convex set C in a dual Banach space X ∗ . Given ε > 0, there is an x ∈ X, with x < ε, such that ϕ + x attains its supremum on C at an extreme point of C. As a consequence, we can easily obtain the following result: Corollary 3.2 Let P ∈ P(N X, Y ). For every ε > 0 there is Q ∈ P(N X, Y ) such that Q < ε and (P + Q)t attains its norm. Proof
Consider the function f : BY ∗ → R given by f (y ∗ ) = P t (y ∗ ) ,
y ∗ ∈ BX ,
which is a w∗ -lower semicontinuous, convex and Lipschitz function. By Theorem 3.1, given ε > 0, there is y0 ∈ Y with y0 < ε and z0∗ ∈ BY ∗ such that f (y ∗ ) + Re y ∗ (y0 ) ≤ f (z0∗ ) + Re z0∗ (y0 ),
∀ y ∗ ∈ BX ∗ .
Then Re z0∗ (y0 ) = z0∗ (y0 ) = |z0∗ (y0 )|. If f (z0∗ ) = 0, then we choose an element x∗0 ∈ SX ∗ and define Q(x) = x∗0 (x)N y0 ,
x ∈ X,
which gives an N -homogeneous polynomial from X into Y satisfying Q = y0 < ε. For any element y ∗ ∈ BY ∗ we have (P + Q)t (y ∗ ) = sup y ∗ (P (x)) + x∗0 (x)N y ∗ (y0 ) x∈BX
≤ P t (y ∗ ) + |y ∗ (y0 )| ≤ P t (z0∗ ) + |z0∗ (y0 )| = |z0∗ (y0 )|, N and (P + Q)t (z0∗ ) = z0∗ (y0 )(x∗0 ) , so (P + Q)t (z0∗ ) = |z0∗ (y0 )|. Hence (P + Q)t attains its norm at z0∗ .
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If f (z0∗ ) = 0, let us define Q(x) =
P t z0∗ (x) y0 , P t z0∗
x ∈ X.
Then it is clear that Q ∈ P(N X, Y ) and Q = y0 < ε. Now we will check that (P + Q)t attains its norm. We do have P + Q =
sup P t (y ∗ ) + Qt (y ∗ ) ≤
y ∗ ∈BY ∗
sup P t (y ∗ ) + |y ∗ (y0 )|
y ∗ ∈BY ∗
≤ P t (z0∗ ) + |z0∗ (y0 )| and
z ∗ (y0 ) |z ∗ (y0 )| (P + Q)t (z0∗ ) = P t (z0∗ ) 1 + 0t ∗ = P t (z0∗ ) 1 + 0t ∗ P (z0 ) P (z0 ) = P t (z0∗ ) + |z0∗ (y0 )|.
Hence (P + Q)t attains its norm at z0∗ , as we wanted to show.
The following result is an improvement of the previous one in the case where Y = M(K) (the space of Radon measures on a compact space K) for a certain class of N -homogeneous polynomials. The proof uses the ideas of Schachermayer [17, Theorem B] who proved a similar result for the linear case. Proposition 3.3 Let X be a Banach space, K a compact Hausdorff space and P : X → M(K) an element in PA (N X, M(K)). Then P can be approximated by approximable N -homogeneous polynomials whose transposes attain their norms at points of C(K). Proof Without loss of generality we may assume that P = 1. Put W := P (BX ). By our assumptions W is a w-compact subset of M(K). Hence W is a Radon–Nikod´ ym set (see [18, Theorem 4] and [3, Theorem II.1]). By using the Stegall optimization principle [5, Theorem 14], there exists a rank one operator T : M(K) → M(K) satisfying T < ε such that the set W1 = (I + T )(W ) has a point μ0 of maximum norm. Now, by using [17, Lemma 3.1] and [19, Lemma 1], we can find a norm one operator S : M(K) → M(K) such that S(μ) − μ < ε for all μ ∈ W and, moreover, there is a function f0 ∈ SC(K) satisfying |Sμ0 (f0 )| = μ0 . Define the N -homogeneous polynomial Q : X → M(K) by Q = S(I + T )P which satisfies that Q − P = sup (SP + ST P − P )(x) x∈BX
≤ sup (S − I)P (x) + ST P x∈BX
≤ sup (S − I)(μ) + ST P μ∈W
≤ ε + T < 2ε.
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We claim that Qt attains its norm at a continuous function. Indeed, we have Q = sup S(I + T )P (x) = sup S(I + T )(μ) x∈BX
μ∈W
≤ sup (I + T )(μ) = μ0 . μ∈W
Moreover Qt (f0 ) = sup |Qt (f0 )(x)| = sup |Q(x)(f0 )| = sup |S(I + T )P (x)(f0 )| x∈BX
x∈BX
x=1
= sup S(I + T )(μ) (f0 ) ≥ |S(μ0 )(f0 )| = μ0 . μ∈W
Hence from these two estimations we obtain Q = μ0 = Qt (f0 ) . Since P ∈ PA (N X, M(K)) and Q = S(I +T )P , we have that Q also belongs to PA (N X, M(K)), and the proof is completed. Corollary 3.4 Let X be a Banach space whose dual has the approximation property, K a compact Hausdorff space and P : X → M(K) a polynomial of Pw (N X, M(K)). Then P can be approximated by polynomials of Pw (N X, M(K)) whose transposes attain their norms at points of C(K). Acknowledgements It is our pleasure to thank Marian Fabian and Rafael Pay´ a for fruitful discussions and helpful remarks.
References [1] Bishop, E., Phelps, R.: A proof that every Banach space is subreflexive. Bull. Amer. Math. Soc., 67, 97–98 (1961) [2] Bourgain, J.: On dentability and the Bishop–Phelps property. Israel J. Math., 28, 265–271 (1977) [3] Bourgain, J.: La propriet´e de Radon–Nikod´ ym, Publications Math´ematiques de L’Universit´e Pierre et Marie Curie, 36, 1979 [4] Stegall, C.: Optimization of functions on certain subsets of Banach spaces. Math. Ann., 236, 171–176 (1978) [5] Stegall, C.: Optimization and differentiation in Banach spaces. Linear Algebra Appl., 84, 191–211 (1986) [6] Poliquin, R. A., Zizler, V. E.: Optimization of convex functions on w∗ -compact sets. Manuscripta Math., 68, 249–270 (1990) [7] Finet, C., Georgiev, P.: Optimization by n-homogeneous polynomial perturbations. Bull. Soc. Roy. Sci. Li` ege, 70, 251–257 (2001) [8] Fabian, M., Zizler, V.: An elementary approach to some questions in higher order smoothness in Banach spaces. Extracta Math., 14, 295–327 (1999) [9] Cheng, L. X., Liu, X. Y., Zuo, M. F.: A linear perturbed Palais–Smale condition for lower semicontinuous functions on Banach spaces. Acta Mathematica Sinica, English Series, 24(11), 1853–1860 (2008) [10] Ferrer, J.: On Banach spaces whose unit sphere determines polynomials. Acta Mathematica Sinica, English Series, 23(1), 175–188 (2007) [11] He, Y. R., Mao, X., Z., Zhou, M.: Strict feasibility of variational inequalities in reflexive Banach spaces. Acta Mathematica Sinica, English Series, 23(3), 563–570 (2007) [12] Fabian, M., Habala, P., H´ ajek, P., Montesinos, V., Pelant, J., Zizler, V.: Functional Analysis and InfiniteDimensional Geometry, CMS Books in Mathematics/Ouvrages de Math´ematiques de la SMC, 8, SpringerVerlag, New York, 2001
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[13] Phelps, R. R.: Convex functions, monotone operators and differentiability. Lecture Notes in Mathematics, 1364, Springer-Verlag, Berlin, 1993 [14] Wu, C., Cheng, L., Ha, M., Lee, E. S.: Convexification of nonconvex functions and application to minimum and maximum principles for nonconvex sets. Comput. Math. Appl., 31(7), 27–36 (1996) [15] Acosta, M. D., Alaminos, J., Garc´ıa, D., Maestre, M.: On holomorphic functions attaining their norms. J. Math. Anal. Appl., 297, 625–644 (2004) [16] Dineen, S.: Complex Analysis on Infinite Dimensional Spaces, Springer Monographs in Mathematics, Springer-Verlag, London, 1999 [17] Schachermayer, W.: Norm attaining operators on some classical Banach spaces. Pacific J. Math., 105, 427–438 (1983) [18] Bourgin, R. D.: Geometric aspects of convex sets with the Radon–Nikod´ ym property, Lecture Notes in Mathematics, 993, Springer-Verlag, Berlin, 1983 [19] Alaminos, J., Choi, Y. S., Kim, S. G., Pay´ a, R.: Norm attaining bilinear forms on spaces of continuous functions. Glasgow Math. J., 40, 359–365 (1998)
