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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1999 • Question 19

Question

Math 6410 • ODE Fall 1999 • Question 19

Let \(H\) be a Hilbert space and let \[ A:D(A)\to H \] be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving necessary and sufficient conditions for the unique solvability of the problem \[ \begin {aligned} u'+Au&=0,\\ u(0)&=u_0\in D(A). \end {aligned} \] Be sure to define pertinent terms.

Raw LaTeX

Let \(H\) be a Hilbert space and let
\[
  A:D(A)\to H
\]
be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving
necessary and sufficient conditions for the unique solvability of the problem
\[
\begin{aligned}
  u'+Au&=0,\\
  u(0)&=u_0\in D(A).
\end{aligned}
\]
Be sure to define pertinent terms.

Topic Tags

Bounded Linear Operators Hilbert Spaces and Orthogonality