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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1997 • Question 3

Question

Math 6420 • PDE Fall 1997 • Question 3

Let \(H\) be a Hilbert space and \(L:H\to H\).

  1. Find and prove the condition that ensures \[ Lu=\int _0^1 k(x,y)u(y)\,dy \] is a bounded linear operator in \(L^2[0,1]\).
  2. Under what condition does the adjoint operator \(L^*\) exist? Assuming that this condition is satisfied, give \(L^*\).

Raw LaTeX

Let \(H\) be a Hilbert space and \(L:H\to H\).
\begin{enumerate}
\item[(a)] Find and prove the condition that ensures
\[
  Lu=\int_0^1 k(x,y)u(y)\,dy
\]
is a bounded linear operator in \(L^2[0,1]\).
\item[(b)] Under what condition does the adjoint operator \(L^*\) exist? Assuming that this
condition is satisfied, give \(L^*\).
\end{enumerate}

Topic Tags

Adjoint Operators and Riesz Representation Bounded Linear Operators Dual Spaces and Bounded Linear Functionals Linear Programming and Optimization