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Báo cáo toán học: "Support functions for matrix ranges: Analogues of Lumer's formula "
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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hỗ trợ chức năng cho các phạm vi ma trận: tương tự của công thức của Lumer. | Copyright by INCREST 1982 J. OPERATOR THEORY 7 1982 25-49 SUPPORT FUNCTIONS FOR MATRIX RANGES ANALOGUES OF LUMER S FORMULA FRANCIS J. NARCOWICH and JOSEPH D. WARD J. INTRODUCTION If sđ and 3 are c -algebras and cTc is the set of complex m X m matrices with identity matrix and identity map 911 - 91i a linear map Ị sđ - ăă is said to be completely positive if the associated maps p fm 9lc ĨS ẩlẽ m 1 are all positive. Stinespring introduced completely positive maps and proved an elegant useful representation theorem for them in 17 . Such maps have recently played a role in classifying c -algebras 8 9 Arveson 2 p. 301 used completely positive maps from a c -algebra sđ with identity I into e lc to define generalized state spaces 1.1 ỗ p sd - 911 I p is completely positive p Z and for T e the matrix ranges 1.2 jy r pei R p p T peSn . He then showed the importance of matrix ranges by proving that if T is a compact irreducible linear operator on a separable Hilbert space then the set Jtyj T Ỉ 1 2 . constitutes a complete set of unitary invariants for T 2 Section 2.5 . Matrix ranges are generalizations of the numerical range indeed W T is the numerical range of T i.e. in the Banach algebra sense it is the closure of the usual Hilbert space numerical range . Each matrix range W T shares with W T the property of being a compact convex subset of a finite dimensional vector space 2 p. 301 which may be viewed as real R2 c for W T and R2 s e ll for Wn T . Such subsets are characterized by their support functions cf. 13 Chapter 13 Section 2 of this paper . 26 FRANCIS J. NARCOWICH and JOSEPH D. WARD The support function for 14zj T is defined by 1.3 ffi - sup Re zm co e W T z e c . Because zWỵ T ỈVỵịỉT 1.3 can be computed via Lumet s formula 12 1.4 Hfz lim i- 7--1 I a J The geometric interpretation of 1.4 is simply that is the directed distance from the origin to the line tangent to WfT with outward normal eiớ. The purpose of this paper is to find formulae for the support function .