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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2015 • Question 4

Question

Math 6420 • PDE Spring 2015 • Question 4

Let \((\lambda _n)\in \ell ^2\) and consider the operator \(T:\ell ^2\to \ell ^2\) defined by \(y=Tx\), where \(x=(\xi _j)\in \ell ^2\), \(y=(\eta _j)\in \ell ^2\) and \[ \eta _j=\sum _{k=1}^{\infty }\alpha _{jk}\xi _k,\qquad j=1,2,\ldots , \] where the \(\alpha _{jk}\) satisfy \[ \sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\lvert \alpha _{jk}\rvert ^2<\infty . \] Show that \(T\) is a compact operator. Hint: Approximate \(T\) with finite rank operators \(T_n\).

Raw LaTeX

Let \((\lambda_n)\in\ell^2\) and consider the operator \(T:\ell^2\to\ell^2\) defined by
\(y=Tx\), where \(x=(\xi_j)\in\ell^2\), \(y=(\eta_j)\in\ell^2\) and
\[
  \eta_j=\sum_{k=1}^{\infty}\alpha_{jk}\xi_k,\qquad j=1,2,\ldots,
\]
where the \(\alpha_{jk}\) satisfy
\[
  \sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\lvert\alpha_{jk}\rvert^2<\infty.
\]
Show that \(T\) is a compact operator. \textbf{Hint:} Approximate \(T\) with finite rank
operators \(T_n\).

Topic Tags

Compact Operators Finite Rank Operators and Approximation Linear Programming and Optimization