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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2021 • Question 2

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2021 • Question 2

Let \(T : H \to H\) be a bounded linear operator on a Hilbert space \(H\). Let \(A\) and \(B\) be subsets of \(H\) such that \(T(A) \subset B\). Show that \(A^\perp \supset T^*(B^\perp )\). Note: We used the notation \(T(A) = \{T x \mid x \in A\}\).

Raw LaTeX

Let \(T : H \to H\) be a bounded linear operator on a Hilbert space \(H\). Let \(A\) and \(B\) be
subsets of \(H\) such that \(T(A) \subset B\). Show that \(A^\perp \supset T^*(B^\perp)\).
Note: We used the notation \(T(A) = \{T x \mid x \in A\}\).

Topic Tags

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