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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1995 • Question 10

Question

Math 6410 • ODE Fall 1995 • Question 10

A bounded linear operator \(P\) in a complex Hilbert space \(\mathcal {H}\) is a projection if and only if \(P^2=P\). Prove that a projection \(P\) is an orthogonal projection (so that \(P\mathcal {H}\perp (1-P)\mathcal {H}\)) if and only if \(P\) is selfadjoint. (You may use the known theorem that a bounded linear operator \(A\) in \(\mathcal {H}\) is selfadjoint if and only if \((Ax,x)\) is real for all \(x\in \mathcal {H}\).)

Raw LaTeX

A bounded linear operator \(P\) in a complex Hilbert space \(\calH\)
is a projection if and only if \(P^2=P\). Prove that a projection \(P\) is an orthogonal
projection (so that \(P\calH\perp(1-P)\calH\)) if and only if \(P\) is selfadjoint. (You
may use the known theorem that a bounded linear operator \(A\) in \(\calH\) is
selfadjoint if and only if \((Ax,x)\) is real for all \(x\in\calH\).)

Topic Tags

Adjoint Operators and Riesz Representation Bounded Linear Operators Convex Sets and Projection Theorems Dual Spaces and Bounded Linear Functionals