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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2017 • Question 3

Question

Math 6420 • PDE Fall 2017 • Question 3

Let \(X\) be an inner product space with inner product \(\langle \cdot ,\cdot \rangle \). Let \((a_j)\) be a sequence of complex numbers and let \((\psi _j)\) and \((\phi _j)\) be two orthonormal sequences in \(X\). Let \(T:X\to X\) be an operator defined by \[ Tx=\sum _{j=1}^{\infty } a_j\langle \psi _j,x\rangle \phi _j. \] State and prove a (nontrivial) sufficient condition on \((a_j)\) so that \(T\) is a compact operator.

Raw LaTeX

Let \(X\) be an inner product space with inner product \(\langle\cdot,\cdot\rangle\).
Let \((a_j)\) be a sequence of complex numbers and let \((\psi_j)\) and \((\phi_j)\) be two
orthonormal sequences in \(X\). Let \(T:X\to X\) be an operator defined by
\[
  Tx=\sum_{j=1}^{\infty} a_j\langle\psi_j,x\rangle\phi_j.
\]
State and prove a (nontrivial) sufficient condition on \((a_j)\) so that \(T\) is a compact
operator.

Topic Tags

Compact Operators Complex Numbers and Elementary Functions Linear Programming and Optimization Orthonormal Bases and Total Sets