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Báo cáo toán học: "Orbits of the unit sphere of ${\calL}\left({{\calX}{\rm{,}}\;{\cal K}}\right)$ under symplectic transformations "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Quỹ đạo của mặt cầu đơn vị $ {\ gọi} \ left ({{\ calx} {\ rm {}} \, {\ cal K}} \ bên phải) Dưới $ biến đổi symplectic. | J. OPERATOR THEORY 11 1984 171-191 Copyright by INCREST 1984 ORBITS OF THE UNIT SPHERE OF UNDER SYMPLECTIC TRANSFORMATIONS N. J. YOUNG 1. INTRODUCTION The set of all biholomorphic transformations of the Open unit disc of the complex plane onto itself comprises the mappings p z --- 1 az where a 1 and Ấ 1. These mappings play an important role in the multiplicative theory of analytic functions and it is not surprising that their analogues are prominent in the extension of this theory to more general domains. One domain for which there is a rich theory is the unit ball A of the space i Xf Jf of all bounded linear operators from xe to X with the operator norm where Xf Xf are Hilbert spaces. We say that W -A - A is biholomorphic if it is Fréchet differentiable on A and has a Fréchet differentiable inverse IP-1 A - A. It then transpires see 4 that every biholomorphic mapping of A onto itself is of the form L i where L is a linear isometry on Xf xf and 0 is a symplectic transformation that is a mapping of the form 1 i X AX B CX D 1 where BeW.J Ce X D e i XT we write . for e xe xf and 2 A Bl-1 c D. - A - c B D These 2x2 matrices of operators represent elements of X XX in an obvious way. The relation 2 is precisely what is needed to ensure that the linear fractional transformation be defined throughout A and map A into itself see 10 . The set of all transformations Ị defined by 1 and 2 is a group. It can also be N. J. YOUNG 172 described as the connected component of the identity in the group of all biholo-morphic transformations of the identity under a certain natural topology 10 . Symplectic transformations arise in many other ways too. A seminal paper by c. L. Siegel 12 inaugurated the study in a number-theoretic spirit of the geometry of the unit ball of the symmetric nxn matrices under this group. They occur naturally in the theory of spaces with indefinite inner product Krein spaces see 8 and hence also in sundry interpolation and approximation problems which are .