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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2022 • Question 3

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2022 • Question 3

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 \((T_k)\) of finite rank operators 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 \((T_k)\) of finite rank operators
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