Defect indices of powers of a contraction

Hwa Long Gau, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Let A be a contraction on a Hilbert space H. The defect index dA of A is, by definition, the dimension of the closure of the range of I - A* A. We prove that (1) dAn ≤ ndA for all n ≥ 0, (2) if, in addition, An converges to 0 in the strong operator topology and dA = 1, then dAn = n for all finite n, 0 ≤ n ≤ dim H, and (3) dA = dA* implies dAn = dAn * for all n ≥ 0. The norm-one index kA of A is defined as sup {n ≥ 0 : {norm of matrix} An {norm of matrix} = 1}. When dim H = m < ∞, a lower bound for kA was obtained before: kA ≥ (m / dA) - 1. We show that the equality holds if and only if either A is unitary or the eigenvalues of A are all in the open unit disc, dA divides m and dAn = ndA for all n, 1 ≤ n ≤ m / dA. We also consider the defect index of f (A) for a finite Blaschke product f and show that df (A) = dAn, where n is the number of zeros of f.

Original languageEnglish
Pages (from-to)2824-2833
Number of pages10
JournalLinear Algebra and Its Applications
Volume432
Issue number11
DOIs
StatePublished - 1 Jun 2010

Keywords

  • Blaschke product
  • Contraction
  • Defect index
  • Norm-one index

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