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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2025 • Question 2

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2025 • Question 2

Consider the operator \(T : L^2[0, 1] \to L^2[0, 1]\) defined by

\[(Tx)(t) = tx(t).\]

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

Raw LaTeX

Consider the operator \(T : L^2[0, 1] \to L^2[0, 1]\) defined by

\[(Tx)(t) = tx(t).\]

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

Topic Tags

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