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

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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

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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

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

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