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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2013 • Question 1

Question

Math 6420 • PDE Fall 2013 • Question 1

Suppose that \(E\) is a bounded linear operator on a Hilbert space \(H\), with operator norm \(\lVert E\rVert =r<1\). Show that the operator \(I-E\), where \(I\) is the identity, is bounded and invertible. Give an estimate for \(\lVert (I-E)^{-1}\rVert \), and a convergent series representation for \((I-E)^{-1}\).

Raw LaTeX

Suppose that \(E\) is a bounded linear operator on a Hilbert space \(H\), with operator norm
\(\lVert E\rVert=r<1\). Show that the operator \(I-E\), where \(I\) is the identity, is bounded
and invertible. Give an estimate for \(\lVert(I-E)^{-1}\rVert\), and a convergent series
representation for \((I-E)^{-1}\).

Topic Tags

Bounded Linear Operators Dual Spaces and Bounded Linear Functionals Linear Programming and Optimization Matrix Norms and Induced Norms