NUMERICAL RADIUS OF A HOLOMORPHIC MAPPING

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1 Geometric Complex Analysis edited by Junjiro Noguchi et al. World Scientific, Singapore, 1995 pp.1 7 NUMERICAL RADIUS OF A HOLOMORPHIC MAPPING YUN SUNG CHOI Department of Mathematics Pohang University of Science and Technology Pohang 790, Korea Let X be a complex Banach space and B be its open unit ball. For each nonnegative integer n, P( n X, X) denotes the Banach space of all continuous n-homogeneous polynomials of X into X, endowed with the norm P = sup{ P (x) : x 1}. In particular, a continous 1-homogeneous polynomial of X into X is a bounded linear operator of X into X. A mapping f : B X is called holomorphic if f is continuous and for each one dimensional affine subspace E of X and for each φ X, φ f is a holomorphic mapping of one complex variable on B E. In this case, there exists a sequence (P n ) of continuous n-homogeneous polynomials such that f(x) = n=0 P n(x) for all x B, which is called the Taylor series expansion for f about 0. G. Lumer [13] has given a theory of the numerical range for bounded linear operators on a Banach space which is a very successful generalization of the classical theory on a Hilbert space. The numerical range for holomorphic mappings was introduced by L. Harris [12], who applied it to his previous results concerning the rotundity at the identity of the sup norm on holomorphic mappings [11, Th. 2] and the extremal case of the Schwarz lemma [10, Th. 1]. Suppose that f : B X is a holomorphic mapping with a continuous extension to the closed unit ball B. We define the set V (f) = {φ(f(x)) : φ = x = 1 = φ(x), φ X, x X} 1

2 2 YUN SUNG CHOI to be the numerical range of f. For a holomorphic mapping f : B X, the numerical radius of f is defined to be the number v(f) = lim sup s 1 (sup{ λ : λ V (f s )}), where f s (x) = f(sx). Clearly if f has a uniformly continuous extension to B, then v(f) = sup{ λ : λ V (f)}, and we say that f attains its numerical radius if there exist φ 0 X and x 0 X such that φ 0 = x 0 = 1 = φ 0 (x 0 ) and v(f) = φ 0 (f(x 0 )). For a holomorphic mapping f : B X with unit radius of boundedness, i.e., with f s = sup{ f s (x) : x B} < for all 0 < s < 1, v(f) = lim sup s 1 v(f s ). The concept of numerical radius seems irrelevant to a holomorphic mapping, because it originated from the study of bounded linear operators. However, L. Harris showed the following inequality in terms of the numerical radius and its nice application as mentioned above. Proposition 1([12]). Let f : B X be holomorphic, and let P n be the n-th term of the Taylor series expansion for f about 0. Then P n k n v(f), where k 0 = 1, k 1 = e and k n = n n/(n 1) for n 2. From this inequality, he proved following his previous results easily. Let δ 0. Suppose that f : B X is a holomor- Proposition 2([11]). phic mapping satisfying for all λ 1. Then P n k n δ. I + λf 1 + δ Proposition 3([10]). Let f : B B be a holomorphic mapping. If Df(0) is an isometry of X onto X, then f is linear, i.e., f is the restriction of Df(0) to B. Motivated by his nice application of the numerical radius of a holomorphic mapping, we will consider the denseness of the set NRA(P( n X, X)) of numerical radius attaining n-homogeneous polynomials of X into X in P( n X, X). In the case of n = 1, i.e., bounded linear operators, after the celebrated paper of E. Bishop and R. Phelps [4], a great deal of attention has been paid to the

3 NUMERICAL RADIUS 3 study of norm attaining operators and in parallel with that, to the study of numerical radius attaining operators. In 1972, B. Sims [15] showed that on a Hilbert space, the set of self-adjoint operators that attain their numerical radii is dense in the set of self-adjoint operators, and raised the question if the set NRA(L(X, X)) of numerical radius attaining operators of X into X is dense in the space L(X, X) of bounded operators of X into X. Since then, several partial results have been given. I. Berg and B. Sims [5] proved the denseness of the set NRA(L(X, X)) for a uniformly convex Banach space. C. Cardassi proved it for a uniformly smooth Banach space [7], the spaces c 0, l 1 [6], C(K) [8], and L 1 (µ) in her unpublished work. M. Acosta and R. Payá proved for a reflexive Banach space [l] and next for a Banach space with the Radon-Nikodym property [3]. In 1992, R. Payá [14] answered the general question to the negative, by exhibiting a Banach space X such that NRA(L(X, X)) is not dense in L(X, X). However, M. Acosta [2] showed in 1993 that every real Banach space can be renormed to satisfy the denseness of the set NRA(L(X, X)). Concerning the denseness of the set NRA(P( n X, X)), our main interest will be the cases X = c 0 and a Banach space with the Radon-Nikodym property. For further details about the following and results on analogous problems related with multilinear mappings, we refer to [9]. We recall property (β) which generalizes the geometric behaviour of the elements {(e n, e n)} in c 0 and l. A Banach space X has property (β) if there exists a subset {(x α, φ α ) : α Λ} of X X satisfying the following assertions: (i) x α = φ α = φ α (x α ) = 1. (ii) There exists λ, 0 λ < 1 such that φ α (x β ) λ for α β. (iii) For every x X, x = sup{ φ α (x) : α Λ}. Let P P( n c 0, c 0 ) with P = 1 and ɛ, 0 < ɛ < 1, be given. For each φ X define P φ P( n c 0 ) by P φ(x) = φ(p (x)), where P( n c 0 ) denotes the Banach space of all continuous complex-valued n-homogeneous polynomials on c 0. Then P = sup P e n = 1, because {(e n, e n)} satisfies the conditions for property (β). Let n 0 be such that P e n 0 1 ɛ/4. Since c 0 has a shrinking basis and the Dunford-Pettis property, every polynomial in P( n c 0 ) can be approximated by norm attaining polynomials in P( n c 0 ), which attain their norms in the closed unit ball of a subspace of c 0 spanned by finitely many usual basis elements (see [9, Theorem 2.2]). From this fact we can choose R P( n c 0 ) so that P e n 0 R ɛ/2, 1 ɛ/4 R 1, and R = R(x ) for some point x in the unit ball of the finite dimensional subspace X 0 spanned

4 4 YUN SUNG CHOI by {e 1,..., e k0 } for some k 0. Put Q(x) = P (x) + [(1 + ɛ)r P e n 0 )(x)]e n0. Then Q P( n c 0, c 0 ) and it is easily checked that P Q 2ɛ and Q attains its norm at x. Let Q j = e j Q for each positive integer j. Since Q(x ) c 0, there exists j 0 such that Q j0 (x ) = Q(x ) = Q. Let l = max{k 0, j 0 }. Let X 1 be the subspace of c 0 spanned by {e 1, e 2,..., e l } and Q 1 = Q X 1, the restriction of Q to X 1. Clearly Q 1 P( n X 1, c 0 ). Define v(q 1 ) = sup{ φ(q 1 (x) : x X 1, φ c 0, x = φ = 1 = φ(x)}. Since X 1 and the subspace of c 0 spanned by the set Q 1 (X 1 ) are finite dimensional, Q 1 attains v(q 1 ). Q = v(q) by Theorem 3.1 in [9], and so v(q 1 ) Q 1 = Q = v(q). Hence if we prove v(q 1 ) = Q 1, then Q attains its numerical radius. Let Q 1 j = e j Q1 for each positive integer j. Then Q 1 j 0 (x ) = Q 1 j 0 = Q 1. By the maximum modulus theorem, there are y S X1, e j 0 (y) = 1 such that Q 1 j 0 (y) = Q 1 j 0. Write y(j 0 ) = e j 0 (y). Note ( ) 1 that sgn(y(j 0 )) y = 1 = e 1 j0 sgn(y(j 0 )) y. In addition, ) v(q 1 ) e j 0 (Q 1 1 ( sgn(y 1 (j 0 )) y = e j 0 (Q 1 (y)) = Q 1 j 0 (y) = Q 1 j 0 = Q 1, and so we have the following. Theorem 4([9]). in P( n c 0, c 0 ). For each positive integer n, NRA(P( n c 0, c 0 )) is dense For x in the unit sphere S of X, let us consider the w -compact subset of X given by D(x) = {ψ X : ψ = ψ(x) = 1}. Note that D(x) by the Hahn-Banach theorem. For P P( n X, X) we define a real function Ψ P on B by Ψ P (x) = max{ ψ(p (x)) : ψ D(x)} if x = 1, Ψ P (x) = x n Ψ P (x/ x ) if 0 < x < 1, Ψ P (0) = 0. Then Ψ P is an upper semicontinuous function on B. In fact, Assume without loss of generality that P = 1, so that 0 Ψ P (x) 1 for all x in B. Let

5 NUMERICAL RADIUS 5 {(x n )} be a sequence in B converging to x 0 and satisfying Ψ P (x n ) r > 0 for all n. Then x n r and x 0 r. Write x n = x n / x n, x 0 = x 0 / x 0. For each n, choose ψ n D( x n ) so that ψ n (P ( x n )) = Ψ P ( x n ). Let ψ 0 be a w -cluster point for the sequence {ψ n } in the closed unit ball of X. Then 1 ψ 0 ( x 0 ) = ψ n ( x n ) ψ 0 ( x 0 ) x n x 0 + ψ n ( x 0 ) ψ 0 ( x 0 ) for all n, and so ψ 0 ( x 0 ) = 1. and so ψ 0 D( x 0 ). Since ψ n (P ( x n )) ψ 0 (P ( x 0 )) P ( x n ) P ( x 0 ) + ψ n (P ( x 0 )) ψ 0 (P ( x 0 )), {ψ n (P ( x n ))} converges to ψ 0 (P ( x 0 )). From ψ n (P ( x n )) = Ψ P ( x n ) = (1/ x n )Ψ P (x n ) r/( x n ), we deduce that Ψ P ( x 0 ) ψ 0 (P ( x 0 )) r/( x 0 ), and so as required. Ψ P (x 0 ) = x 0 Ψ P ( x 0 ) r, Recall that a closed bounded convex set E in X is called a Radon-Nikodym set if for any closed subset C of E, the set of functionals that strongly expose C is norm-dense in X. A bounded above function f : C R strongly exposes C if f attains its maximum at a point x of C, and any sequence {x k } of points in C satisfying {f(x k )} f(x) converges to x. Proposition 5([16]). Let f be a bounded above, upper semicontinuous function on a Radon-Nikodym set E of a real Banach space X. Then the set is a G δ -dense subset of X. {φ X : f + φ strongly exposes E} Applying Proposition 5 and the fact that Ψ P is an upper semicontinuous function on B, which is a Radon-Nikodym set for a Banach space X with the Radon-Nikodym property, given P P( n X, X) and ɛ > 0, there is φ X with 0 < φ < ɛ such that Ψ P + Re φ strongly exposes B. In particular, there is x 0 B such that Ψ P (x 0 )+Re φ(x 0 ) Ψ P (x)+re φ(x) for all x B. It follows by rotating x that Ψ P (x 0 )+φ(x 0 ) Ψ P (x)+ φ(x) for all x B and x 0 = 1.

6 6 YUN SUNG CHOI Choose ψ 0 X so that ψ 0 = 1 = ψ 0 (x 0 ) and ψ 0 (P (x 0 )) = Ψ P (x 0 ). Define a polynomial Q P( n X, X) by Q(x) = P (x) + λφ(x)ψ 0 (x) n 1 x 0 (x X) with λ = 1, ψ 0 (P (x 0 )) = λ ψ 0 (P (x 0 )). Then P Q < ɛ, and it is easily checked that ψ(q(x)) ψ 0 (Q(x 0 )) for each x, x = 1 and each ψ D(x), and so we have the following. Theorem 6([9]). Suppose that X has the Radon-Nikodym property. Then for each positive integer n, NRA(P( n X, X)) is dense in P( n X, X). References [ 1 ] M. D. Acosta, Denseness of operators whose second adjoints attain their numerical radii, Proceddings Amer. Math. Soc., 105 (1989), [ 2 ] M. D. Acosta, Every real Banach space can be renormed to satisfy the denseness of numerical radius attaining operators, Israel J. Math., 81 (1993), [ 3 ] M. D. Acosta and R. Payá, Numerical radius attaining operators and the Radon- Nikodym property, Bull. London Math. Soc., 25 (1993), [ 4 ] E. Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc., 67 (1961), [ 5 ] I. D. Berg and B. Sims, Denseness of numerical radius attaining operators, J. Austral. Math. Soc. Ser. A, 36 (1984), [ 6 ] C. S. Cardassi, Numerical radius attaining operators, Banach spaces, Proceedings Missouri 1984, Lecture Notes in Math., 1166 (ed. N. Kalton and E. Saab, Springer, Berlin, 1985), [ 7 ] C. S. Cardassi, Density of numerical radius attaining operators on some reflexive spaces, Bull. Austral. Math. Soc., 31 (1985), 1-3. [ 8 ] C. S. Cardassi, Numerical radius attaining operators on C(K), Proc. Amer. Math. Soc., 95 (1985), [ 9 ] Y. S. Choi and S. G. Kim, Norm or numerical radius attaining multilinear mappings and polynomials, J. London Math. Soc. (to appear). [10] L. A. Harris, Schwarz s lemma in normed spaces, Proceedings of the National Academy of Sciences, U.S.A., 62 (1969),

7 NUMERICAL RADIUS 7 [11] L. A. Harris, A continuous form of Schwarz s lemma in normed linear spaces, Pacific J. Math., 38 (1971), [12] L. A. Harris, The numerical range of holomorphic functions in Banach spaces, Amer. J. Math., 43 (1971), [13] G. Lumer, Semi-inner-product spaces, Transactions Amer. Math. Soc., 100 (1961), [14] R. Payá, A counterexample on numerical radius attaining operators, Israel J. Math., 79 (1992), [15] B. Sims, On numerical range and its application to Banach algebras, Ph.D. Dissertation, University of Newcastle, Australia (1972). [16] C. Stegall, Optimization and differentiation in Banach spaces, Linear Algebra Appl., 84 (1986),