Question

Exercise [4] Let T∈ B(H) be an invertible self-adjoint operator. Show that ||T^-1|| = inf{[λ] : λ E ∈ O(T)}^-1

Answer 1

Given T ∈ B(H) is an invertible self-adjoint operator. To show that ||T-1||=inf{λ:[λ]∈O(T)}-1We know that ∥T-1∥ ≥ inf{λ:[λ]∈O(T)}-1... (1) Now, let ε > 0 be given. By definition of spectral radius, there exists λ ∈ O(T) such that [λ] > ||T||-1. Now, consider x ∈ H such that ∥x∥ = 1 and Tx = λx. Then (T-λ)x = 0 ⇒ ∥(T-λ)x∥ = 0 ⇒ ∥Tx-λx∥ = 0 ⇒ ∥Tx∥ = ∥λx∥. Therefore, ∥Tx∥ = |λ| = [λ].So, [λ] ≥ ||T||-1+ε. Hence, ||T-1|| ≤ [λ]-1 ≤ (||T||-1+ε)-1. Since ε is arbitrary, we get ||T-1|| ≤ inf{λ:[λ]∈O(T)}-1... (2) From (1) and (2), we can conclude that ∥T-1∥ = inf{λ:[λ]∈O(T)}-1.