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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2021 • Question 1

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2021 • Question 1

Let \(H\) be a Hilbert space endowed with the inner product \(\langle \cdot , \cdot \rangle \). Let \(T\) be a bounded linear operator satisfying \(\langle T x, x\rangle \ge 0\) for all \(x \in H\). Show that \(I + T\) is injective.

Raw LaTeX

Let \(H\) be a Hilbert space endowed with the inner product \(\langle \cdot, \cdot\rangle\). Let \(T\) be a bounded
linear operator satisfying \(\langle T x, x\rangle \ge 0\) for all \(x \in H\). Show that \(I + T\) is injective.

Topic Tags

Bounded Linear Operators Dual Spaces and Bounded Linear Functionals Hilbert Spaces and Orthogonality