Numerical radius inequality for C0 contractions

Pei Yuan Wu, Hwa Long Gau, Ming Cheng Tsai

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

We show that if A is a C0 contraction with minimal function φ{symbol} such that w (A) = w (S (φ{symbol})), where w (·) denotes the numerical radius of an operator and S (φ{symbol}) is the compression of the shift on H2 ⊖ φ{symbol} H2, and B commutes with A, then w (AB) ≤ w (A) {norm of matrix} B {norm of matrix}. This is in contrast to the known fact that if A = S (φ{symbol}) (even on a finite-dimensional space) and B commutes with A, then w (AB) ≤ {norm of matrix} A {norm of matrix} w (B) is not necessarily true. As a consequence, we have w (AB) ≤ w (A) {norm of matrix} B {norm of matrix} for any quadratic operator A and any B commuting with A.

Original languageEnglish
Pages (from-to)1509-1516
Number of pages8
JournalLinear Algebra and Its Applications
Volume430
Issue number5-6
DOIs
StatePublished - 1 Mar 2009

Keywords

  • Compression of the shift
  • Numerical radius
  • Numerical range
  • Quadratic operator

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