臺灣博碩士論文加值系統

摘要:

這篇論文討論兩個問題：

(i).
研究作用在矩陣代數上的一類線性映射，它們在一個小集合上保持譜。

(ii).
我們應用 Choi-Kraus 定理，以研究正規矩陣 (算子) 之間的完全正插值問題。我們給出其存在性的充份必要條件，這些條件依賴於
數值域的包含關係和矩陣擴張問題。

In this thesis, we will consider the following problems:

(i).
We study linear maps of matrix algebras, which preserve some spectral functions on a small
subset of mn x mn matrices.

(ii).
We study completely positive interpolations between normal matrices (operators) using
their spectrum and the Choi-Kraus form. We obtain a necessary and sufficient condition
for the existence of a completely positive interpolation in terms of conditions about
numerical ranges and dilations.

Keywords:

目 錄
論文審定書…………………………………………………………… i
論文誌謝……………………………………………………………… ii
中文摘要………………………………………………………….….. iii
Abstract ………………………………………..……………………. iv

Contents
Chapter 1: Introduction 1
1.1 Linear maps preserving spectral properties of tensor products of matrices . . 1
1.2 Completely positive interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Chapter 2: Preliminaries 7
2.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Banach spaces and Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Linear operators on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Spectra of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Numerical ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 Completely positive maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.8 Compact and Schatten-p class operators . . . . . . . . . . . . . . . . . . . . 16
2.9 A technical lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter 3: Linear maps preserving spectral properties of tensor product
of low rank matrices 20
3.1 Spectrum and numerical range preservers . . . . . . . . . . . . . . . . . . . 20
3.2 Spectral radius preservers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Numerical radius preservers . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Chapter 4: Completely Positive Interpolation 34
4.1 Interpolations and numerical range inclusions . . . . . . . . . . . . . . . . . 34
4.2 Completely positive interpolations preserving approximate units or trace . . . 47
4.3 Completely positive interpolations between commutative families . . . . . . 53
4.4 Extension to the General Case . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Computational considerations . . . . . . . . . . . . . . . . . . . . . . . . . 64
Bibliography 68

[1] P. Albinia, A. Toigob, and V. Umanitac, Relations between convergence rates in Schatten
p-norms, J. Math. Phys., 49 (2008), 013504.
[2] T. Ando, Structure of operators with numerical radius one, Acta Sci. Math. (Szeged), 34
(1973), 11-15.
[3] W. B. Arveson, Subalgebras of C -algebras, Acta Math., 123 (1969), 141-224.
[4] M. S. Bazaraa, J. J. Jarvis, and H. D. Sherali, Linear Programming and Network Flows,
Wiley, 1990
[5] V. P. Belavkin and P. Staszewski, Radon-Nikodym theorem for completely positive
maps, Reports on Mathematical Physics, 1 (1986), 49-55.
[6] J. Bergh and J. L¨ofstr¨om, Interpolation Spaces: An Introduction, Springer Verlag, 1976.
[7] D. P. Bertsekas, Convex Analysis and Optimization, Athena Scientific, 2003.
[8] Q. Bu, C. Liao, and N. C. Wong, Orthogonally additive and orthogonally multiplicative
holomorphic functions of matrices, Ann. Funct. Anal. 5, 2, (2014).
[9] C. Carath´eodory, Uber den Variabilitatsbereich der Fourierschen Konstanten von positiven
harmonischen Funktionen, Rendiconti del Circolo Matematico di Palermo 32
(1911), 193V217.
[10] W. S. Cheung, C. K. Li, and Y. T. Poon, Isometries Between Matrix Algebras, J. Aust.
Math. Soc. 77 (2004), 1-16.
[11] M. D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl.
10 (1975), 285-290.
[12] M. D. Choi and C. K. Li, Numerical ranges and dilations, Linear and Multilinear Algebra
47 (2000), 35-48.
[13] M. D. Choi and C. K. Li, Constrained unitary dilations and numerical ranges, J. Operator
Theory 46 (2001), 435-447.
[14] J. B. Conway A Course in Functional Analysis, Second ed., Springer, New York, 1990.
[15] A. Fosner, Z. Huang, C.K. Li, Y.T. Poon, and N.S. Sze, Linear maps preserving the
higher numerical ranges of tensor product of matrices, Linear and Multilinear Algebra,
to appear.
[16] A. Fosner, Z. Huang, C.K. Li, and N.S. Sze, Linear preservers and quantum information
science, Linear and Multilinear Algebra, (2012), 1-14.
[17] A. Fosner, Z. Huang, C.K. Li, and N.S. Sze, Linear maps preserving Ky Fan norms
and Schatten norms of tensor product of matrices, SIAM J. Matrix Analysis Appl., 34,
(2013), 673-685.
[18] A. Fosner, Z. Huang, C.K. Li, and N.S. Sze, Linear maps preserving numerical radius
of tensor product of matrices, J. Math. Anal. Appl., 407 (2013), 183-189.
[19] G. Frobenius, Uber die Darstellung der endlichen Gruppen durch lineare Substitutionen,
Sitzungsber. Deutsch. Akad. Wiss. Berlin, (1897), 994-1015.
[20] P. R. Halmos, A Hilbert Space Problem Book, Second ed., Springer, New York, 1982.
[21] I. N. Herstein, Topics in Ring Theory, U. of Chicago Lecture Notes, 1965.
[22] F. Kittaneh, Inequalities for the Schatten p-norm II, Glasgow Math. J. 29 (1987), 99-104.
[23] K. Kraus, Effects and Operations: Fundamental Notions of Quantum Theory, Springer
Verlag, 1983.
[24] C. K. Li and S. Pierce, Linear preserver problems, Amer. Math. Monthly, 108 (2001),
591-605.
[25] C. K. Li and Y. T. Poon, Interpolation by completely positive maps, Linear Multilinear
Alg. 59 (2011), 1159-1170.
[26] C. K. Li, Y. T. Poon, and N. S. Sze, Linear preservers of tensor product of unitary orbits,
and product numerical range, Linear Algebra Appl., 438:3797-3803, (2013)
[27] M. Marcus and B. N. Moyls, Linear transformations on algebras of matrices, Canadian
J. Math., 11:61-66, (1959).
[28] M. Marcus and B. N. Moyls, Transformations on tensor product spaces, Pacific J. Math.,
9:1215-1221, (1959).
[29] B. A. Mirman, Numerical range and norm of a linear operator, Voronez. Gos. Univ. Trudy
Sem. Funkcional Anal., 10 (1968), 51-55.
[30] L. Moln ar, Selected preserver problems on algebraic structures of linear operators and
on function spaces, Lecture Notes in Mathematics, 1895. Springer-Verlag, Berlin, 2007.
[31] M. A. Nielsen and I. L. Chuang, Quantum Comptation and Quantum Information, Cambridge
Univeristy Press, 2000.
[32] V. Pellegrini, Numerical range preserving operators on a Banach algebra, Linear Algebra
Appl., 54:143-147, (1975).
[33] V. I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University
Press, 2002.
[34] F. M. Pollack, Numerical range and convex sets, Canad. Math. Bull. 17 (1974), 295-296.
[35] B. Simon, Analysis with weak trace ideals and the number of bound states of
Schrodinger operators, Trans. Amer. Math. Soc., 224 (1976), 367-380.
[36] B. Simon, Trace Ideals and Their Applications , Cambridge University Press, 1979.
[37] Ł. Skowronek, E. Størmer, and K. ˙ Zyczkowski, Cones of positive maps and their duality
relations, J. Math. Phys. 50 (2009), 062106.
[38] W. F. Steinespring, Positive functions on C -algebras, Proc. Amer. Math. Soc., 6 (1955),
211-216.
[39] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, American
Elsevier, New York, 1970.
[40] R. Westwick, Transformations on tensor spaces, Pacific J. Math., 23:613-620, (1967).
[41] G. Wittstock, Ein operatorwertiger Hahn-Banach satz, J. Funct. Anal., 40 (1981), 127-
150.