Defect indices of powers of a contraction

Hwa Long Gau, Pei Yuan Wu

研究成果: 雜誌貢獻期刊論文同行評審

5 引文 斯高帕斯(Scopus)


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.

頁(從 - 到)2824-2833
期刊Linear Algebra and Its Applications
出版狀態已出版 - 1 6月 2010


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