Defect indices of powers of a contraction |
| |
Authors: | Hwa-Long Gau Pei Yuan Wu |
| |
Institution: | a Department of Mathematics, National Central University, Chung-Li 320, Taiwan b Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan |
| |
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?dimH, and (3) dA=dA∗ implies dAn=dAn∗ for all n?0. The norm-one index kA of A is defined as sup{n?0:‖An‖=1}. When dimH=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. |
| |
Keywords: | 15A03 47A30 |
本文献已被 ScienceDirect 等数据库收录! |
|