Extend the numerical-radius gap bound (28) to all powers

Establish that for every bounded linear operator T on a Hilbert space X and every integer n ≥ 1, the inequality w(T) − w(T^n) ≤ ||T||^2 − c(T)^2 holds, where w(T) denotes the numerical radius of T and c(T) denotes the Crawford number of T.

Background

In Example 3.5, the paper derives a new bound on the gap between the numerical radius of a bounded linear operator T and the numerical radius of its square T^2, namely w(T) − w(T^2) ≤ ||T||^2 − c(T)^2, where ||T|| is the operator norm and c(T) is the Crawford number. This estimate is obtained using an enhanced Buzano-type inequality developed earlier in the paper.

It is classically known that w(T^n) ≤ w(T) for all n ≥ 1, so the gap w(T) − w(T^n) is nonnegative. The conjecture seeks to generalize the derived bound from n = 2 to arbitrary powers n, providing a uniform control of the gap by a quantity depending only on T (namely ||T||^2 − c(T)^2). Such bounds are relevant in operator theory and applications, including stability analysis of numerical schemes and diagnostics in Markov chain Monte Carlo.