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Self Adjoint Operators

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Find the adjoint differential operator L* and the space on which it acts if:

Lu = u"+au'+bu

Where
u(0)=u'(1)
u(1)=u'(0)

Lu = -[p(x)u']'+q(x)u

Where
u(0)=u(1)
u'(0)=u('1)

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https://brainmass.com/math/manifolds/self-adjoint-operators-18120

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The expert examines self adjoint operators. The space of the acts are given functions are provided.

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The Eigenvectors of a Self-Adjoint (Hermitian)

Please see the attachment for the full problem description and hint.

Let T >= 0 be a strictly positive definite linear operator on a finite dimensional inner product space V over F = R or C.

(a) Prove that the exponential map Exp: A -> e^A = sum infinity k = 0 1/kl A^k is one-to-one from the space of self-adjoint operators H = {T : T* = T} into the set of positive definite operators

P = {T : T* = T and (Tx,x) >= 0 for all x E V}

In particular e^A is self-adjoint if A is self-adjoint.

(b) Prove that the exponential map Exp is a surjective map from H to P.

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