MINKOWSKI PRODUCT OF CONVEX SETS AND PRODUCT NUMERICAL RANGE

被引：2

作者：

Li, Chi-Kwong ^{[1
]}

Pelejo, Diane Christine ^{[1
]}

Poon, Yiu-Tung ^{[2
]}

Wang, Kuo-Zhong ^{[3
]}

机构：

[1] Coll William & Mary, Dept Math, Williamsburg, VA 23187 USA

[2] Iowa State Univ, Dept Math, Ames, IA 50011 USA

[3] Natl Chiao Tung Univ, Dept Appl Math, Hsinchu 30010, Taiwan

来源：

OPERATORS AND MATRICES | 2016年 / 10卷 / 04期

基金：

美国国家科学基金会;

关键词：

Convex sets; Minkowski product; numerical range;

D O I：

10.7153/oam-10-53

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Let K-1, K-2 be two compact convex sets in C. Their Minkowski product is the set K1K2 = {ab : a is an element of K-1, b is an element of K-2}. We show that the set K1K2 is star-haped if K-1 is a line segment or a circular disk. Examples for K-1 and K-2 are given so that K-1 and K-2 are triangles (including interior) and K1K2 is not star-shaped. This gives a negative answer to a conjecture by Puchala et. al concerning the product numerical range in the study of quantum information science. Additional results and open problems are presented.

引用

页码：945 / 965

页数：21

共 8 条

[1]

ANDO T, 1973, ACTA SCI MATH, V34, P11

[2] Numerical ranges and dilations [J].

Choi, MD ;

Li, CK .

LINEAR & MULTILINEAR ALGEBRA, 2000, 47 (01) :35-48

[3]

Farouki R. T., 2002, Reliable Computing, V8, P43, DOI 10.1023/A:1014737602641

[4] Minkowski geometric algebra of complex sets [J].

Farouki, RT ;

Moon, HP ;

Ravani, B .

GEOMETRIAE DEDICATA, 2001, 85 (1-3) :283-315

[5]

Horn RA., 2013, MATRIX ANAL

[6]

LI C. K., 1986, LINEAR MULTILINEAR A, V20, P5

[7] PRODUCTS OF SETS OF COMPLEX NUMBERS [J].

MCALLISTER, BL .

TWO-YEAR COLLEGE MATHEMATICS JOURNAL, 1983, 14 (05) :390-397

[8] Product numerical range in a space with tensor product structure [J].

Puchala, Zbigniew ;

Gawron, Piotr ;

Miszczak, Jaroslaw Adam ;

Skowronek, Lukasz ;

Choi, Man-Duen ;

Zyczkowski, Karol .

LINEAR ALGEBRA AND ITS APPLICATIONS, 2011, 434 (01) :327-342

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