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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2009 • Question 4

Question

Math 6420 • PDE Fall 2009 • Question 4

Let \(T:\ell ^2\longrightarrow \ell ^2\) be defined by \(y=Tx\) with \(y_j=x_je^{-j}\) for \(j=1,2,3,\ldots \), where \(x=(x_1,x_2,\ldots )\) and \(y=(y_1,y_2,\ldots )\). Prove that \(T\) is a compact operator on \(\ell ^2\).

Raw LaTeX

Let \(T:\ell^2\longrightarrow\ell^2\) be defined by \(y=Tx\) with
\(y_j=x_je^{-j}\) for \(j=1,2,3,\ldots\), where
\(x=(x_1,x_2,\ldots)\) and \(y=(y_1,y_2,\ldots)\). Prove that \(T\) is a compact operator on
\(\ell^2\).

Topic Tags

Compact Operators Linear Programming and Optimization