TY - JOUR

T1 - A generalized spectral radius formula and Olsen's question

AU - Loring, Terry

AU - Shulman, Tatiana

PY - 2012

Y1 - 2012

N2 - Let A be a C⁎C⁎-algebra and I be a closed ideal in A. For x∈Ax∈A, its image in A/IA/I is denoted by x˙, and its spectral radius is denoted by r(x)r(x). We prove that max{r(x),‖x˙‖}=inf‖(1+i)−1x(1+i)‖ (where the infimum is taken over all i∈Ii∈I such that 1+i1+i is invertible), which generalizes the spectral radius formula of Murphy and West. Moreover if r(x)<‖x˙‖ then the infimum is attained. A similar result is proved for a commuting family of elements of a C⁎C⁎-algebra. Using this we give a partial answer to an open question of C. Olsen: if p is a polynomial then for “almost every” operator T∈B(H)T∈B(H) there is a compact perturbation T+KT+K of T such that ‖p(T+K)‖=‖p(T)e‖‖p(T+K)‖=‖p(T)‖e.

AB - Let A be a C⁎C⁎-algebra and I be a closed ideal in A. For x∈Ax∈A, its image in A/IA/I is denoted by x˙, and its spectral radius is denoted by r(x)r(x). We prove that max{r(x),‖x˙‖}=inf‖(1+i)−1x(1+i)‖ (where the infimum is taken over all i∈Ii∈I such that 1+i1+i is invertible), which generalizes the spectral radius formula of Murphy and West. Moreover if r(x)<‖x˙‖ then the infimum is attained. A similar result is proved for a commuting family of elements of a C⁎C⁎-algebra. Using this we give a partial answer to an open question of C. Olsen: if p is a polynomial then for “almost every” operator T∈B(H)T∈B(H) there is a compact perturbation T+KT+K of T such that ‖p(T+K)‖=‖p(T)e‖‖p(T+K)‖=‖p(T)‖e.

U2 - 10.1016/j.jfa.2011.10.005

DO - 10.1016/j.jfa.2011.10.005

M3 - Journal article

VL - 262

SP - 719

EP - 731

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -