7SURJECTIVE ISOMETRIES ON A BANACHSPACE OF ANALYTIC FUNCTIONSON THE OPEN UNIT DISC

日本大学・薬学部 丹羽典朗
NORIO NIWA, SCHOOL OF PHARMACY,NIHON UNIVERSITYThis work was supported by the Research Institute for Mathematical

Sciences, a Joint Usage/Research Center located in Kyoto University.
1. INTRODUCTION

Let (M, \Vert\cdot\Vert_{M}) and (N, \Vert\cdot\Vert_{N}) be normed linear spaces, respectively.
A mapping T:(M, \Vert\cdot\Vert_{M})arrow(N, \Vert\cdot\Vert_{N}) is an isometry if and only if it
preserves the distance of two points in M , that is,\Vert T(a)-T(b)\Vert_{N}=\Vert a-b\Vert_{M} (a, b\in M) .
Here, we assume that T is not necessarily complex linear. The Mazur‐

Ulam theorem [16] states that every surjective isometry T between two
normed linear spaces is real linear provided T(0)=0.

We mention the characterization of isometries on several normed

linear spaces. Isometries were studied on various spaces by many re‐searchers, as for example in [3, 12, 13, 21, 22]. In 1932, isometries arestudied by Banach [1, Theorem 3 in Chapter XI] (see also [24, Theo‐rem 83]). There have been numerous papers on isometries defined on
Banach spaces of analytic functions; see [2, 4, 5, 8, 11, 14].

Among the basic problems in analytic function spaces, Novinger andOberlin, in [20], characterized complex linear isometries on a normed

space S^{p} . The underlying space S^{p} is a normed space consisting ofanalytic functions f on the open unit disc \mathbb{D} whose derivative f' belongs

to the classical Hardy space (H^{p}(\mathbb{D}), \Vert\cdot\Vert_{p}) for 1\leq p<\infty .
introduced the norm |f(0)|+\Vert f'\Vert_{p} on the normed space S^{p}.

TheyIn this talk, we study surjective isometries on the Banach space S_{A}of analytic functions f defined on \mathbb{D} whose derivative can be extended

to the closed unit disc \overline{\mathb {D} , and endowed with the norm \Vert f\Vert_{\sigma}=|f(0)|+\sup_{z\in \mathbb{D}}|f'(z)| . We denote by A(\overline{\mathb {D} ) the disc algebra, that is, the algebraof all analytic functions on \mathbb{D} which can be extended to continuousfunctions on \overline{\mathb {D} .

8
2. MAIN RESULTLet A(\overline{\mathb {D} ) be the Banach space of all analytic functions on the open
unit disc \mathbb{D} that can be continuously extended to the closed unit disc\overline{D} with the supremum norm on \mathbb{D} . For each v\in A(\overline{\mathbb{D} ), v' means thederivative of v on \mathbb{D} , that is,v'(z)= \lim_{harrow 0}\frac{v(z+h)-v(z)}{h} (z\in \mathbb{D}).

We define S_{A} by the linear space of all analytic functions f on \mathbb{D} whose
derivative f' belongs to A(\overline{\mathb {D} ) . By [6, Theorem 3.11], we see that S_{A}\subset
A(\overline{\mathb {D} ) . By the definition of S_{A}, f' is an analytic function on \mathbb{D} which
can be extended to a continuous function on \overline{\mathb {D} . Let \hat{v} be the uniquecontinuous extension of v\in A(\overline{\mathbb{D} ) to \overline{\mathb {D} . In fact, such an extension isunique since \mathbb{D} is dense in \overline{\mathb {D} . We define the norm \Vert f\Vert_{\sigma} of f\in S_{A} by\Vert f\Vert_{\sigma}=|f(0)|+\Vert\hat{f'}\Vert_{\infty} (f\in S_{A}) ,where \Vert\hat{f'}\Vert_{\infty}=\sup\{|\hat{f'}(z)| : z\in\overline{\mathbb{D}}\}= \sup\{|f'(z)| : z\in \mathbb{D}\} . It is not(2.1)

difficult to check that (S_{A}, \Vert\cdot\Vert_{\sigma}) is a complex Banach space.

Theorem 1. If T:(S_{A}, \Vert\cdot\Vert_{\sigma})arrow(S_{A}, \Vert\cdot\Vert_{\sigma}) is a surjective, not neces‐sarily complex linear, isometry, then one of the following four forms isoccured;
there exist constants c_{1,1}, c_{1,2}, \lambda_{1}\in \mathbb{T} and a_{1}\in \mathbb{D} such that

T(f)(z)=T(0)(z)+c_{1,1}f(0)+ \int_{[0,z]}c_{1,2}f'(\rho(\zeta))d\zeta (\forall f\in S_{A}, \forall z\in \mathbb{D}),there exist constants c_{2,1}, c_{2,2}, \lambda_{2}\in \mathbb{T} and a_{2}\in \mathbb{D} such that

T(f)(z)=T(0)(z)+ \overline{c_{2,1}f(0)}+\int_{[0,z]}c_{2,2}f'(\rho(\zeta) d\zeta (\forall f\in S_{A}, \forall z\in \mathbb{D}),there exist constants c_{3,1}, c_{3,2}, \lambda_{3}\in \mathbb{T} and a_{3}\in \mathbb{D} such that

T(f)(z)=T(0)(z)+c_{3,1}f(0)+ \int_{[0,z]}\overline{c_{3,2} f'(\rho(\overline{\zeta}) }d\zeta (\forall f\in S_{A}, \forall z\in \mathbb{D})there exist constants c_{4,1}, c_{4,2}, \lambda_{4}\in \mathbb{T} and a_{4}\in \mathbb{D} such that

T(f)(z)=T(0)(z)+ \overline{c_{4,1}f(0)}+\int_{[0,z]}\overline{c_{4,2} f'(\rho(\overline{\zeta}) }d\zeta (\forall f\in S_{A}, \forall z\in \mathbb{D})for all z\in\overline{\mathbb{D} and for j=1,2,3,4.
\rho(z)=\lambda_{j}\frac{za_{j}{\overline{},a_{j}
z1}
Conversely, each of the above forms is a surjective isometry on S_{A}

wherewith the norm \Vert\cdot\Vert_{\sigma} , where T(0) is an arbitrary element of S_{A}.

,,9We start by defining an embedding of S_{A} into a subspace B consistingof complex valued continuous functions. Then using the Arens‐Kelleytheorem (see [10, Corollary 2.3.6 and Theorem 2.3.8]), we give a char‐acterization of extreme points of the unit ball B_{1}^{*} of the dual space B^{*}of B . Then we construct some maps to describe extreme points of B_{1}^{*}.
We used an idea by Ellis for the characterization of surjective real

linear isometries on uniform algebras (see [9]). An adjoint operator of asurjective real linear isometry on the dual space B^{*} preserves extremepoints. The action of such adjoint operator on the set of extreme points
gives a representation for the isometries on B . We show that the isome‐tries of S_{A} are integral operators of weighted differential operators.

For the details of proof, refer to [18].
REFERENCES[1] S. Banach, Theory of linear operations, Translated by F. Jellett, Dover Publi‐cations, Inc. Mineola, New York, 2009.

[2] F. Botelho, Isometries and Hermitian operators on Zygmund spaces, Canad.
Math. Bull. 58 (2015), 241‐249.
[3] M. Cambern, Isometries of certain Banach algebras, Studia Math. 25 (1964‐1965) 217‐225.
[4] J.A. Cima and W.R. Wogen, On isometries of the Bloch space, Illinois J. Math.
24 (1980), 313‐316.
[5] K. deLeeuw, W. Rudin and J. Wermer, The isometries of some function spaces,
Proc. Amer. Math. Soc. 11 (1960), 694‐698.
[6] P.L. Duren, The theory of H^{p} spaces, Academic Press, New York, 1970
[7] F. Forelli, The isometries of H^{p} , Canad. J. Math. 16 (1964), 721‐728.
[8] F. Forelli, A theorem on isometreis and the application of it to the isometriesof H^{p}(S) for 2<p<\infty , Canad. J. Math. 25 (1973), 284−289[9] A.J. Ellis, Real characterizations of function algebras amongst function spaces,
Bull. London Math. Soc. 22 (1990), 381‐385.
[10] R. Fleming and J. Jamison, Isometries on Banach spaces: function spaces,Chapman & Hal1/CRC Monographs and Surveys in Pure and Applied Math‐ematics, 129. Chapman & Hal1/CRC , Boca Raton, FL, 2003.
[11] W. Hornor and J.E. Jamison, Isometreis of some Banach spaces of analytic
functions, Integral Equations Operator Theory 41 (2001), 410‐425.
[12] K. Jarosz and V.D. Pathak, Isometries between function spaces, Trans. Amer.
Math. Soc. 305 (1988), 193‐205.
[13] K. Kawamura, H. Koshimizu and T. Miura, Norms on C^{1}([0,1]) and theirisometries, Acta Sci. Math. (Szeged) 84 (2018), 239‐261.
[14] C.J. Kolaski, Isometries of Bergman spaces over bounded Runge domains,Canad. J. Math. 33 (1981), 1157‐1164.
[15] H. Koshimizu, Linear isometries on spaces of continuously differentiable andLipschitz continuous functions, Nihonkai Math. J. 22 (2011), 73‐90.
[16] S. Mazur and S. Ulam, Sur les transformationes isométriques d’espaces vecto‐riels normés, C. R. Acad. Sci. Paris 194 (1932), 946‐948.

10[17] T. Miura, Surjective isometries between function spaces, Contemp. Math. 645(2015), 231‐239.
[18] T. Miura and N. Niwa, Surjective isometries on a Banach space of analytic
functions on the open unit disc, Nihonkai Math. J. 29 (2018), 53‐67.
[19] M. Nagasawa, Isomorphisms between commutative Banach algebras with anapplication to rings of analytic functions, Ko‐dai Math. Sem. Rep. 11 (1959),182‐188.

[20] W.P. Novinger and D.M. Oberlin, Linear isometries of some normed spaces ofanalytic functions, Canad. J. Math. 37 (1985), 62‐74.
[21] V.D. Pathak, Isometries of C^{(n)} [0,1], Pacific J. Math. 94 (1981), 211‐222.
[22] N.V. Rao and A.K. Roy, Linear isometries of some function spaces, Pacific J.
Math. 38 (1971), 177‐192.
[23] W. Rudin, Real and complex analysis. Third edition. McGraw‐Hill Book Co.,New York, 1987.

[24] M.H. Stone, Applications of the theory of Boolean rings to general topology,
Trans. Amer. Math. Soc. 41 (1937), 375‐481.