JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (6): 92-98.doi: 10.6040/j.issn.1671-9352.0.2022.485

Previous Articles     Next Articles

The numerical range of the operator P+QP

Chen ZHANG(),Weiyan YU*()   

  1. College of Mathematics and Statistics, Hainan Normal University, Haikou 571158, Hainan, China
  • Received:2022-09-16 Online:2023-06-20 Published:2023-05-23
  • Contact: Weiyan YU E-mail:2449311935@qq.com;wyyume65@163.com

RichHTML

5

PDF (PC)

174

Abstract:

In this paper, the numerical range of the combination P+QP of two orthogonal projections P and Q on a complex separable Hilbert space $\mathscr{H}$ is studied. Firstly, the support function of the numerical range of P+QP is given by using the method of operator block. Next, by using the properties of the support function, a geometric characterization of the numerical range of the operator P+QP is given. That is, the closure of its numerical range is a closed convex hull of some explicit ellipses parameterized by points in the spectrum.

Key words: numerical range, support function, orthogonal projections

CLC Number: 

  • O177.1

Fig.1

Ellipses $\mathscr{E}(\lambda)$ for λ=1.1, 1.2, ⋯, 1.9"

Fig.2

$\overline{\operatorname{conv}\left\{\cup_{\lambda \in[1, 2]} \mathscr{E}(\lambda)\right\}}$"

1 STONE M H . Linear transformations in Hilbert space and their applications to analysis[M]. American: American Mathematical Society, 1932.
2 TOEPLITZ O . Dasalgebraische analogon zu einem satze von fejér[J]. Mathematische Zeitschrift, 1918, 2 (1/2): 187- 197.
3 GUSTAFSON K E , RAO D K . Numerical range[M]. New York: Springer, 1997.
4 HALMOS P R . Two subspaces[J]. Transactions of the American Mathematical Society, 1969, 144, 381- 389.
doi: 10.1090/S0002-9947-1969-0251519-5
5 KLAJA H . The numerical range and the spectrum of a product of two orthogonal projections[J]. Journal of Mathematical Analysis and Applications, 2014, 411 (1): 177- 195.
doi: 10.1016/j.jmaa.2013.09.024
6 DU Hongke , LI Chikwong , WANG Kuozhong , et al. Numerical ranges of the product of operators[J]. Operators and Matrices, 2017, 11, 171- 180.
7 WANG Yueqing , ZUO Ning , DU Hongke . Characterizations of the support function of the numerical range of the product of positive contractions[J]. Linear and Multilinear Algebra, 2016, 64 (10): 2068- 2080.
doi: 10.1080/03081087.2016.1138439
8 LENARD A . The numerical range of a pair of projections[J]. Journal of Functional Analysis, 1972, 10 (4): 410- 423.
doi: 10.1016/0022-1236(72)90037-7
9 GERYBA T , SPITKOVSKY I M . On some 4-by-4 matrices with bi-elliptical numerical ranges[J]. Linear and Multilinear Algebra, 2021, 69 (5): 855- 870.
doi: 10.1080/03081087.2020.1857326
10 GERYBA T , SPITKOVSKY I M . On the numerical range of some block matrices with scalar diagonal blocks[J]. Linear and Multilinear Algebra, 2021, 69 (5): 772- 785.
doi: 10.1080/03081087.2020.1749225
11 CHAN J T, LI C K, POON Y T. Commuting normal operators and joint numericalrange[EB/OL]. 2021: arXiv: 2108.05414[math. FA]. https://arxiv.org/abs/2108.05414.
12 BÖTTCHER A . A gentle guide to the basics of two projections theory[J]. Linear Algebra and Its Applications, 2010, 432 (6): 1412- 1459.
doi: 10.1016/j.laa.2009.11.002
13 邓春源, 杜鸿科. 两子空间的公共补与Groβ问题[J]. 数学学报(中文版), 2006, 49 (5): 1099- 1112.
DENG Chunyuan , DU Hongke . Common complements of two subspaces and an answer to Groβ's question[J]. Acta Mathematica Sinica(Chinese Series), 2006, 49 (5): 1099- 1112.
14 ROCKAFELLAR R T . Convex analysis[M]. Princeton: Princeton University Press, 1970.
15 RIESZ F , NAGY S B . Functional analysis[M]. Massachusetts: Courier Corporation, 1990.
[1] DUAN Chen-xia1, SUN Gang2, WANG Gui-jun1*. Universal approximation of three-layer regular fuzzy neural networks for a class of functions [J]. J4, 2012, 47(3): 81-86.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] HE Hai-lun， CHEN Xiu-lan* . Circular dichroism detection of the effects of denaturants and buffers on the conformation of cold-adapted protease MCP-01 and  mesophilic protease BP01[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2013, 48(1): 23 -29 .
[2] SUN Xiao-ting1, JIN Lan2*. Application of DOSY in oligosaccharide mixture analysis[J]. J4, 2013, 48(1): 43 -45 .
[3] YANG Ying, JIANG Long*, SUO Xin-li. Choquet integral representation of premium functional and related properties on capacity space[J]. J4, 2013, 48(1): 78 -82 .
[4] Ming-Chit Liu. THE TWO GOLDBACH CONJECTURES[J]. J4, 2013, 48(2): 1 -14 .
[5] ZHAO Tong-xin1, LIU Lin-de1*, ZHANG Li1, PAN Cheng-chen2, JIA Xing-jun1. Pollinators and pollen polymorphism of  Wisteria sinensis (Sims) Sweet[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 1 -5 .
[6] WANG Kai-rong, GAO Pei-ting. Two mixed conjugate gradient methods based on DY[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(6): 16 -23 .
[7] YANG Jun. Characterization and structural control of metalbased nanomaterials[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2013, 48(1): 1 -22 .
[8] DONG Wei-wei. A new method of DEA efficiency ranking for decision making units with independent subsystems[J]. J4, 2013, 48(1): 89 -92 .
[9] ZHANG Jing-you, ZHANG Pei-ai, ZHONG Hai-ping. The application of evolutionary graph theory in the design of knowledge-based enterprises’ organization strucure[J]. J4, 2013, 48(1): 107 -110 .
[10] ZHAO Jun1, ZHAO Jing2, FAN Ting-jun1*, YUAN Wen-peng1,3, ZHANG Zheng1, CONG Ri-shan1. Purification and anti-tumor activity examination of water-soluble asterosaponin from Asterias rollestoni Bell[J]. J4, 2013, 48(1): 30 -35 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd