Acta mathematica scientia,Series A ›› 2020, Vol. 40 ›› Issue (6): 1420-1430.

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The Quadratic Numerical Range and the Spectrum of Some Unbounded Block Operator Matrices

Wenwen Qiu,Yaru Qi*()   

  1. Department of Mathematics, College of Sciences, Inner Mongolia University of Technology, Hohhot 010051
  • Received:2019-10-24 Online:2020-12-26 Published:2020-12-29
  • Contact: Yaru Qi E-mail:qiyaru@imut.edu.cn
  • Supported by:
    the NSFC(11601249)

Abstract:

In this paper, we study the operator M=[0IAB] which is associated with the second order differential equation ¨z(t)B˙z(t)+Az(t)=0 in a Hilbert space, where A is a self-adjoint and uniformly positive linear operator, B is accretive. We prove that M is a boundedly invertible closed operator and ¯M|H1×H1=M where H1=D(A) with the norm xH1=Ax. And we characterize the spectral distribution of the operator M by using the quadratic numerical range of the block operator matrix M|H1×H1.

Key words: Unbounded operator, Block operator matrices, Spectrum, Quadratic numerical range

CLC Number: 

  • O177.1
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