Spectral Radius of a Normal Operator

Abstract

Let X be a Complex Banach space and T be a bounded operator in X. The number sup {|λ| : λ ∈ σ(T)} (where σ(T) is the spectrum of T and σ(T) ̸= ϕ) is called the spectral radius of T and denoted by r(T). Since λ ≤ ∥T∥for all λ ∈ σ(T), it follows that r(T) ≤ ∥T∥. The spectral mapping theorem implies that r(Tn) = (r(T))n for every positive integer n. It frequently turns out that it is easy to compute the spectral radius of an operator even if it is hard to _nd the spectrum. This is often made easy by the spectral radius formula. Let H be a Hilbert space and T be a bounded linear operator in H. In this paper we show that if T is normal, then Tn is normal for each n ∈ N and ∥Tn∥ = ∥T∥n. Consequently, we use the spectral radius formula to show that r(T) = ∥T∥. Moreover, we show that if X is a Complex Banach space and T is bounded in X then there is a λ belonging to the spectrum of T such that |λ| = r(T). Let H be a Complex Hilbert space and T be a bounded operator in H which is normal; we show that ∥T∥ = sup {|Tx, x| : x ∈ H and ∥x∥ = 1} and the residual spectrum of T is void.