Adjoint Of An Operator Puns

So, I'm studying for a linear algebra prelim, and I have a question. I've seen adjoints treated differently in a number of different places, so I just wondered if any of you could help clarify this: How is the conjugate transpose of the matrix form of a linear operator related to its adjoint? What is the relationship between normal matrices and adjoints?

OK.

I know that self-adjoint complex operators always have real eigenvalues. So measurements will always produce a real number which is desirable.

Also Stone's theorem. https://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitar

But this doesn't quite add up to an intuitive understanding yet.

My question is very simple. Given the adjoint of L (not to be confused with the formal adjoint of L), which in turn is given by the formal adjoint L+ such that the bilinear concomitant associated with L is 0, it is evident that if L acting on some function, call it φ, is a non homogeneous boundary value problem, say L[φ] = h(ζ), φ(ζ ) has a solution if, the adjoint exists, then there must be some function, ξ(ζ) that satisfies the boundary conditions that are a result of the bilinear concomitant associated with L being zero if and only if, the inner product of ξ(ζ) with h(ζ), (of course with respect to some weight function, call it ψ(ζ)), is zero; that is to say, given ξ(ζ) and h(ζ) are orthogonal with respect to ψ(ζ), there exists a solution to the non homogeneous, linear differential equation given by Lφ. Obviously this solution is not clearly evident and may not even have a closed form, with that being said, could one not simply use a modified Fourier-Hankel method to solve the system coupled of differential equations given by Lφ and L'𝜏 to obtain a solution by choosing the appropriate weight function, ψ(ζ), such that the inner product of h(ζ) and the inner product of h'(ζ) has an analytic solution? It seems obvious to me that this would yield a much more useful and applicable result.

Addendum: I know this is fairly mathematical, but this is an elementary problem that I feel not only has obvious repercussions in applied fields, but could be understood and intuited by any modern human.

I only know that ||T|| = ||T* || but this equality could hold for a smaller subspace D(T*) of H. Could someone clarify this? Thanks!

Any cool consequences? Any particular reasons why it's so cool?

I have a self-adjoint differential operator (w'''' + n^2 * w'' = 0) with self-adjoint boundary conditions (w(0) = w'(0) = w(1) = w'(1) = 0). If my knowledge of the theory serves me right, this is a Hermitian operator and, thus, the eigenfunctions corresponding to different eigenvalues should be orthogonal.

The eigenfunctions can be divided into two groups: Even (n_i = 2π, 4π, 6π, ...) where w_i(x) = A_i*(1 - cos(n_i *x))

Odd (n_i = 2.86π, 4.92π, 6.94π, ...) where w_i(x) = A_i*(1 - cos(n_i x) - (2/n_i)(n_i *x - sin(n_i *x))

Orthogonality holds between an even and an odd eigenfunction, but not between two even or two odd. Am I incorrect in assuming orthogonality is guaranteed or is something else at play?

This is from Friedberg's Linear Algebra 4th edition, section 6.3 #3c.

V = P_1 (R) with the inner product int(fg) from -1 to 1, and T(f) = f' + 3f. We're meant to evaluate T* with f(t) = 4 - 2t. The answer in the book is 12 + 6t.

I first tried transforming the basis vectors into (3, 0) and (1, 3), forming the matrix

[3   0]
[1   3]

then transposing it and multiplying (4, -2). I get 12 -2t, which doesn't agree with the answer in the book.

I then tried solving directly from the definition of the adjoint to solve for T* in general with x = 1 + x and y = y_1 + y_2 t:

<T(x), y> = <x, T\*(y)>

which gives me the integral of 16 + 4t - 6t^2 , which I can't factor into <1 + x, something> to get T*.

I tried mapping into R^2 (at this point I was just trying anything) and solving from the definition, and I get:

<(w, x), T\*(y, z)> = <T(w, x), (y, z)>
= <(4w, 3x), (y, z)>
= 4wy + 3xz
= <(w, x), (4w, 3x)>

So T* would be 4x + 3xt, which for 4 - 2t gives 16 - 6t, which isn't even what I got before.

I'm completely stumped. I've been working on this one problem for hours now. This is homework so I would appreciate it if you didn't solve the problem for me but just gave a hint about what the hell I'm doing wrong.

Hello!

I am having trouble understanding what an adjoint operator is in the context of differential operators. Near as I can find it, if you have uL(v), where L is your operator, then your adjoint is vL(u)?

I've tried looking at texts but I can't seem to grasp exactly how to find the adjoint. Any help would be appreciated.

I know they generalize the conjugate transpose of a matrix but I am not sure how. I also know they involve a restriction mapping. I have seen some examples of them but I don't understand why the examples satisfy the definition.

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