Just define any, real function

I know a way to define any function (say, a (x)) and then apply an assumption to it (say, a (x) is real). My code will be inserted below.


    A(x_) = Function(x, a(x))
    Assumptions(A(x) (Element) Reals

    In(216):= Integrate(x A(x), {x, -1, 1}) // FullSimplify
    Out(216)= 0

However, it becomes very "chaotic". When I try to define operators for arbitrary functions, the output is as follows:

    Function(x, a(x))

Instead of:

    a(x)

Is there a better way?

If what I just wrote is not detailed enough, here is the exact code:

    In(197):= Clear("Global`*")

    In(198):= (*-v- Arbitrary Real functions*)

    In(199):= (Alpha)(x_) = Function(x, a(x))

    Out(199)= Function(x, a(x))

    In(200):= (Beta)(x_) = Function(x, b(x));

    In(201):= (Gamma)(x_) = Function(x, cc(x));

    In(202):= (Rho)(x_) = Function(x, r(x));

    In(203):= f(n_, x_) = Function({x, n}, ff(x, n));

    In(204):= F(n_, x_, y_) = E^(I n y) f(n, x);

    In(205):= (*-^- Arbitrary Real functions*)

    In(206):= (*-v- Differential Operators*)

    In(207):= (Lambda)n(n_, 
       x_) = ((Alpha)(x)*D(#, {x, 1}) + (Beta)(x) + n*(Rho)(x)) &;

    In(208):= (Mu)n(n_, 
       x_) = ((Alpha)(x)*D(#, {x, 1}) + (Gamma)(x) - n*(Rho)(x)) &;

    In(220):= 
    Tp(x_, y_) = (E^(
     I y) ((Alpha)(x)*D(#, {x, 1}) + (Beta)(x) - 
       I (Rho)(x)*D(#, {y, 1}))) &;

    In(210):= 
    Tm(x_, y_) = (-E^(-I y) ((Alpha)(x)*D(#, {x, 1}) + (Gamma)(x) + 
       I (Rho)(x)*D(#, {y, 1}))) &;

    In(211):= T0(y_) = (-I D(#, {y, 1})) &;

    In(212):= (*-^- Differential Operators*)

    In(213):= (*-v- Assumptions*)

    In(214):= 
    Assumptions((Alpha)(x) (Element) Reals, (Beta)(x) (Element) 
      Reals, (Gamma)(x) (Element) Reals, (Rho)(x) (Element) Reals, 
     F(n, x) (Element) Complexes);

    In(215):= (*-^- Assumptions*)

    In(217):= Tm(x, y)(Tp(x, y)(F(n, x, y))) // FullSimplify

    Out(217)= -E^(-I y) Function(x, cc(x)) + 
     Function(x, b(x)) Function(x, r(x)) + 
     E^(I n y) (-(Function(x, a(x))^2 - 
            Function(x, a(x)) Function(x, 
              r(x)) + (1 + I) (1 + (1 + I) n) Function(x, 
              r(x))^2) Function({x, n}, 0) + 
        n (1 + n) Function(x, r(x))^2 Function({x, n}, ff(x, n))) - 
     Function(x, 
       0) (Function(x, a(x)) + I Function(x, r(x))) (1 + 
        E^(I n y) ((1 - I) Function({x, n}, 0) + 
           n Function({x, n}, ff(x, n))))
```