TY - JOUR

T1 - Defect indices of powers of a contraction

AU - Gau, Hwa Long

AU - Wu, Pei Yuan

N1 - Funding Information:
Corresponding author. E-mail addresses: hlgau@math.ncu.edu.tw (H.-L. Gau), pywu@math.nctu.edu.tw (P.Y. Wu). 1 Research supported by the National Science Council of the Republic of China under NSC 97-2115-M-008-014. 2 Research supported by the National Science Council of the Republic of China under NSC 96-2115-M-009-013-MY3 and by the MOE-ATU project.

PY - 2010/6/1

Y1 - 2010/6/1

N2 - 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.

AB - 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.

KW - Blaschke product

KW - Contraction

KW - Defect index

KW - Norm-one index

UR - http://www.scopus.com/inward/record.url?scp=77949654800&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2009.12.024

DO - 10.1016/j.laa.2009.12.024

M3 - 期刊論文

AN - SCOPUS:77949654800

SN - 0024-3795

VL - 432

SP - 2824

EP - 2833

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 11

ER -