## Answers

If T is self adjoint then < Tx, y > = < x, Ty > where: T: V rightarrow V and x, y elementof V < p, q > = integral^1_- 1 p(t) q(t) dt To check that L is self adjoint, we check that < Lp(t), q(t) > = < p(t), Lq(t) > a) L p(t) = t p(t) Now, < L p(t), q(t) > = integral^1_- 1 Lp(t) q(t) dt = integral^1_- 1 t p(t) q(1) dt = integral^1_- 1 p(t) t q(t) = integral^1_- 1 p(t) L q(t) dt = < p(t), Lq(t) > Hence, L is self adjoint b) Lp(t) = -p(2 - t) Now, < L p(t), q(t) > = integral^1_- 1 Lp(t) q(t) dt = integral^1_- 1 - p(2 - t) q(t) dt Let t = -x Rightarrow dt = -dx at t = 1, x = -1, t = -1 Rightarrow x = 1 \r\n Therefore < Lp(t), q(t) > = integral^1_- 1 - p(x) q(- x) (- dx) = integral^1- 1 p(x) q(- x) dx = integral^1_- 1 p(x) (- q(- x)) dx

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