Let H be a Hilbert space and A a positive self-adjoint operator on H. Prove thatthe following assertions are equivalent:(i) A(H) is dense in H.(ii) Ker A = {0}.(iii) A is positive definite, i.e., (Ax,x) > 0, Vx € H\ {0}.

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Answered: Let H be a Hilbert space and A a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let H be a Hilbert space and A a positive self-adjoint operator on H. Prove that the following assertions are equivalent: (i) A(H) is dense in H. (ii) Ker A = {0}. (iii) A is positive definite, i.e., (Ax, x) > 0, Vx H\ {0}.