Problem 2: Spectral Radius and Operator Norm in Banach Spaces Background: Let X = C([0, 1]), the Banach space of continuous real-valued functions on [0,1] with the sup norm. Define the weighted shift operator T: X→ X by where w(x)=2+ sin(x). Tasks: (Tf)(x) = w(x)f(2), a) Boundedness and Operator Norm: • Show that T is a bounded linear operator on X. • Compute the operator norm ||T||. • Plot Requirement: Plot the weight function w(x) over [0,1]. b) Spectral Radius Calculation: • Determine the spectral radius r(T) of the operator T. • Hint: Use the Gelfand formula and properties of the weighted shift operator. c) Spectrum Visualization: • Describe the spectrum σ (T) of T in the complex plane. • Plot Requirement: Provide a detailed plot of σ(T), highlighting any regions where the spectrum accumulates. d) Comparison with Operator Norm: • Compare the spectral radius r(T) with the operator norm ||T||. Determine whether r(T) < |||T||, r(T) = ||T||, or r(T) > ||T||. Analysis: Discuss the implications of this comparison in the context of the spectral properties of T. e) Perturbation and Stability: Introduce a perturbation T = T+8S, where S is another bounded operator on X defined by (Sf)(x) = f(x), and & is a small scalar. • . Analyze how the spectrum o(Ts) changes as & varies in [-0.05, 0.05]. Plot Requirement: Create a series of plots showing the evolution of σ(Ts) in the complex plane for selected values of 8. Comment on the stability of the spectrum under such perturbations.