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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2015 • Question 3

Question

Math 6420 • PDE Spring 2015 • Question 3

Let \(u,v\) be non-zero elements of a Hilbert space \(H\). Consider the linear operator \(T:H\to H\) defined for \(x\in H\) by \(Tx=u(x,v)\).

  1. Briefly explain why the operator \(T\) is compact.
  2. Find a condition on \(u\) and \(v\) that guarantees that the equation \[ (I-T)x=y \] admits a unique solution \(x\) for all \(y\in H\).

Raw LaTeX

Let \(u,v\) be non-zero elements of a Hilbert space \(H\). Consider the linear operator
\(T:H\to H\) defined for \(x\in H\) by \(Tx=u(x,v)\).
\begin{enumerate}
\item[(a)] Briefly explain why the operator \(T\) is compact.

\item[(b)] Find a condition on \(u\) and \(v\) that guarantees that the equation
\[
  (I-T)x=y
\]
admits a unique solution \(x\) for all \(y\in H\).
\end{enumerate}

Topic Tags

Bounded Linear Operators Hilbert Spaces and Orthogonality Linear Programming and Optimization Zero-Stability and Root Conditions