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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Fall 2023 • Question 4

Question

Math 6710 • Applied Linear Operators and Spectral Methods Fall 2023 • Question 4

Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying

\[\sum _{n=1}^{\infty } \|T e_n\| < \infty .\]

Show that \(T\) is compact by approximating \(T\) by a sequence of finite rank operators \((T_k)\) that converges to \(T\) in an appropriate sense.

Raw LaTeX

Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\).
Let \(T : H \to H\) be a bounded linear operator satisfying

\[\sum_{n=1}^{\infty} \|T e_n\| < \infty.\]


Show that \(T\) is compact by approximating \(T\) by a sequence of finite rank operators \((T_k)\)
that converges to \(T\) in an appropriate sense.

Topic Tags

Bounded Linear Operators Dual Spaces and Bounded Linear Functionals Finite Rank Operators and Approximation Orthonormal Bases and Total Sets