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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2015 • Question 3

Question

Math 6210 • Real Analysis Spring 2015 • Question 3

Recall that if \(A\) is a linear operator on Hilbert space with bounded operator norm, then it has an adjoint \(A^*\) which is the unique operator so that for all \(v,w\), \[ \langle Av,w\rangle =\langle v,A^*w\rangle . \]

(a) Let \(P:\ell ^2(\mathbb N)\to \ell ^2(\mathbb N)\) by \(P\bar v=\bar w\) where \(w_j=0\) if \(j\) is not prime and \(w_j=v_j\) otherwise. What is \(P^*\)?

(b) Let \(P:\ell ^2(\mathbb N)\to \ell ^2(\mathbb N)\) by \(P\bar v=\bar w\) where \(w_j=v_{p_j}\), where \(p_j\) is the \(j\)th prime. What is \(P^*\)?

Raw LaTeX

Recall that if \(A\) is a linear operator on Hilbert space with
bounded operator norm, then it has an adjoint \(A^*\) which is the unique
operator so that for all \(v,w\),
\[
\langle Av,w\rangle=\langle v,A^*w\rangle.
\]

(a) Let \(P:\ell^2(\mathbb N)\to\ell^2(\mathbb N)\) by \(P\bar v=\bar w\) where
\(w_j=0\) if \(j\) is not prime and \(w_j=v_j\) otherwise. What is \(P^*\)?

(b) Let \(P:\ell^2(\mathbb N)\to\ell^2(\mathbb N)\) by \(P\bar v=\bar w\) where
\(w_j=v_{p_j}\), where \(p_j\) is the \(j\)th prime. What is \(P^*\)?

Topic Tags

Bounded Linear Operators Hilbert Spaces and Orthogonality Lp Spaces and Sequence Spaces