spectral radius

The spectral radius answers a simple question: how far out does the spectrum reach? It is the radius of the smallest disk centered at the origin that contains the entire spectrum. It controls long-term growth: roughly, applying the operator many times amplifies by a factor governed by this radius, which is why it appears in stability analysis and iteration schemes.

For a bounded operator T on a complex Banach space, the spectral radius is r(T) = sup{ |lambda| : lambda in sigma(T) }. It is bounded above by the operator norm, r(T) <= ||T||, but can be strictly smaller. The fundamental Gelfand formula gives r(T) = lim of ||T^n||^{1/n} as n -> infinity (the limit always exists), tying a spectral quantity to the growth of powers.

For a normal operator (in particular self-adjoint or unitary) equality holds: r(T) = ||T||. But in general the inequality is strict; a nonzero nilpotent matrix N with N^2 = 0 has spectrum {0}, so r(N) = 0 while ||N|| > 0. So a small spectral radius does NOT mean a small norm, only that high powers eventually shrink in the n-th-root sense.