Unitary part of a contraction

Hwa Long Gau, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


For a contraction A on a Hilbert space H, we define the index j (A) (resp., k (A)) as the smallest nonnegative integer j (resp., k) such that ker (I - Aj * Aj) (resp., ker (I - Ak * Ak) ∩ ker (I - Ak Ak *)) equals the subspace of H on which the unitary part of A acts. We show that if n = dim H < ∞, then j (A) ≤ n (resp., k (A) ≤ ⌈ n / 2 ⌉), and the equality holds if and only if A is of class Sn (resp., one of the three conditions is true: (1) A is of class Sn, (2) n is even and A is completely nonunitary with {norm of matrix} An - 2 {norm of matrix} = 1 and {norm of matrix} An - 1 {norm of matrix} < 1, and (3) n is even and A = U ⊕ A, where U is unitary on a one-dimensional space and A is of class Sn - 1).

Original languageEnglish
Pages (from-to)700-705
Number of pages6
JournalJournal of Mathematical Analysis and Applications
Issue number2
StatePublished - 15 Jun 2010


  • Completely nonunitary part
  • Contraction
  • Norm-one index
  • S-operator
  • Unitary part


Dive into the research topics of 'Unitary part of a contraction'. Together they form a unique fingerprint.

Cite this