Linear maps preserving the higher numerical ranges of tensor products of matrices

Ajda Fošner, Zejun Huang, Chi Kwong Li, Yiu Tung Poon, Nung Sing Sze

Research output: Journal article publicationJournal articleAcademic researchpeer-review

6 Citations (Scopus)

Abstract

For a positive integer n, let Mnbe the set of n × n complex matrices. Suppose m, n ≥ 2 are positive integers and k ∈ {1,..., mn - 1}. Denote by Wk(X) the k-numerical range of a matrix X ∈ Mmn. It is shown that a linear map Φ: Mmn→ Mmnsatisfies (Formula presented.) for all A ∈ Mmand B ∈ Mnif and only if there is a unitary U ∈ Mmnsuch that one of the following holds. (Formula presented.) where (1) φ is the identity map A ⊗ B {mapping} A ⊗ B or the transposition map A ⊗ B {mapping} (A⊗B)t, or (2) min{m, n} ≤ 2 and φ has the form A⊗B {mapping} A⊗Btor A⊗B {mapping} At⊗ B.
Original languageEnglish
Pages (from-to)776-791
Number of pages16
JournalLinear and Multilinear Algebra
Volume62
Issue number6
DOIs
Publication statusPublished - 1 Jan 2014

Keywords

  • Hermitian matrix
  • k-numerical range
  • linear preserver
  • tensor product of matrices

ASJC Scopus subject areas

  • Algebra and Number Theory

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