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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2024 • Question 1

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2024 • Question 1

Let \((\lambda _j)\) be a sequence that is dense in \([0, 1]\) and consider the linear operator \(T : \ell ^2 \to \ell ^2\) defined by its action on a sequence \((\xi _j) \in \ell ^2\) by

\[T(\xi _j) = (\lambda _j \xi _j).\]

Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent \(\rho (T)\).

Raw LaTeX

Let \((\lambda_j)\) be a sequence that is dense in \([0, 1]\) and consider the linear operator \(T : \ell^2 \to \ell^2\)
defined by its action on a sequence \((\xi_j) \in \ell^2\) by

\[T(\xi_j) = (\lambda_j \xi_j).\]


Find the spectrum \(\sigma(T)\), point spectrum \(\sigma_p(T)\), continuum spectrum \(\sigma_c(T)\), residual
spectrum \(\sigma_r(T)\) and resolvent \(\rho(T)\).

Topic Tags

Bounded Linear Operators Dual Spaces and Bounded Linear Functionals Point, Continuous, and Residual Spectrum Spectral Theory of Operators