The Isometries of H ∞ (K)

Abstract

Let K be a finite-dimensional Hubert space. In this article a characterization is given of the linear isometries of the Banach space Hn (K) onto itself. It is shown that T is such an isom- etry iff Tis of the form (TF)(z)=3~F(t(z)), for F e //»(iC) and z belonging to the unit disc, where t is a conformai map of the disc onto itself and 9 is an isometry of K onto K. 0. Introduction. Throughout this paper the letter K represents a finite-dimensional complex Hubert space. We denote by (■, •) the inner product in K, and fix some orthonormal basis {ex, ■ ■ ■ , eN} of K. Let Hœ(K) be the Banach space of functions F defined on the unit circle to K such that the scalar function (F, e) belongs to H°° of the circle for each eeK, and such that ||/:'||a)=ess sup ||F(eia:)|| is finite. (Here ||-||œ denotes the norm in HX(K), and ||-|| that in K.) If FeHx(K), we define the /7°° coordinate functions/, by fn(eix)= (F(etx),en). Then almost everywhere we have 2n=i l/n(e!iI)l2<00» and F(eix)=ILifn(eix)en. Moreover, each Fe Hm(K) may be extended (via a power series) to an analytic function F(z) on the unit disc D={z; \z < 1}, having boundary values a.e. which determine Fon the circle. This analytic function coincides with the function obtained by extending to D, in the usual way, the coordinate functions in the expression F= 2 fnen- Thus, whenever it is convenient to do so, we may think of Hœ(K) as a space of bounded, vector-valued, analytic functions defined on D. In recent years considerable work has been directed toward the deter- mination of what properties of the Hardy classes Hv, l^p^ao, can be generalized to the analogous spaces HV(K) of vector-valued functions. An excellent account of what had been done along these lines through the year 1964 can be found in the book by Helson (2). Here we investigate the isometries of //(AT), which have been described for //°° (i.e. for one- dimensional K), by deLeeuw, Rudin, and Wermer (5), and quite inde- pendently by Nagasawa (6). Although our results generalize those of (5) and (6), the proofs, of necessity, require a quite different approach, since