In the meanwhile you might have figured it out, but nevertheless

(* re-create the identity (A.37): The point is keeping the m invariant, i.e. using identities (A.35) and (A.36)
for D[LegendreP[l,m,Cos[\[CurlyTheta]]],\[CurlyTheta]] together with 
(A.20) and (A.21) for Sin[\[CurlyTheta]] D[LegendreP[l,m,Cos[\[CurlyTheta]]] to do that. 
Did not work out, so use SphericalHarmonicY as an index function as before. *)
Off[SphericalHarmonicY::argrx]
Clear[kp, km, \[Xi], singleQ, \[Zeta]]
km[l_, m_] := Sqrt[(l + m) (l - m)/((2 l + 1) (2 l - 1))]
kp[l_, m_] := km[l + 1, m]
singleQ[G_, f_] := Length[Select[Level[f, {1}], Head[#] == G &]] == 1 /; Head[f] === Times
singleQ[G_, f_] :=  And @@ (singleQ[G, #] & /@ (List @@ f)) /; Head[f] === Plus
singleQ[G_, f_] := False /; Head[f] =!= Times && Head[f] =!= Plus
\[Zeta][G_, y_, f_] := ((D[f,  y] /. {G[l_, m_, x__] ->  kp[l, m] G[l + 1, m, x] + km[l, m] G[l - 1, m, x]}) + 
    (f /. {G[l_, m_, x__] -> -(l/r) kp[l, m] G[l + 1, m, x] + (l + 1)/r km[l, m] G[l - 1, m, x]})) /; ! FreeQ[f, G] 
    && FreeQ[Denominator[Together[f]], G] && singleQ[G, Expand[f]] && FreeQ[D[f, y], Derivative[__][G]]


\[Zeta][SphericalHarmonicY, r, 
  f[r] SphericalHarmonicY[\[Lambda], \[Mu], \[Alpha], \[Beta]]] // TraditionalForm



Simplify[\[Zeta][SphericalHarmonicY, 
   r, \[Zeta][SphericalHarmonicY, r, 
    f[r] SphericalHarmonicY[\[Lambda], \[Mu], \[Alpha], \[Beta]]]]] // TraditionalForm



Simplify[\[Zeta][SphericalHarmonicY, 
   r, \[Zeta][SphericalHarmonicY, 
    r, \[Zeta][SphericalHarmonicY, r, 
     f[r] SphericalHarmonicY[\[Lambda], \[Mu], \[Alpha], \[Beta]]]]]] // TraditionalForm

this last one gives

I still didn't look in the numerical check for the identity, but I am pretty sure the identity is correct. In any case, I will check this identity taken from the book by Bethe and Salpeter, "Quantum Mechanics of One and Two Electron Systems". Even if the formula I wrote has a typo the question how to define an operator that "knows" how to read the indices of the function (e.g. Spherical Harmonic) he acts on is relevant to me.

I'm sorry but I'm still having problems to verify the original relation

It doesn't work out in general, so it was tried to work it out per coefficient of f[r],D[f[r],r] where it didn't work out again. Finally the coefficients have been checked numerically, again unsuccessful:

 In[47]:= FullSimplify[Coefficient[
      Sqrt[((l + m + 1) (l - m + 1))/((2 l + 3) (2 l + 
                1))] SphericalHarmonicY[l + 1, 
           m, \[Theta], \[Phi]] (f'[r] - l f[r]/r) + 
           Sqrt[((l + m) (l - m))/((2 l + 1) (2 l - 1))] (f'[
             r] + (l + 1) f[r]/r) - (Cos[\[Theta]] D[#, r] - 
           Sin[\[Theta]]/r D[#, \[Theta]]) &[f[r] SphericalHarmonicY[l, m, \[Theta], \[Phi]]], f[r]], 
     Element[l | m, Integers]]

    Out[47]= (1/r)((1 + l) Sqrt[((l - m) (l + m))/(-1 + 4 l^2)] + 
      m Cos[\[Theta]] SphericalHarmonicY[l, m, \[Theta], \[Phi]] + (
      E^(-I \[Phi]) Sqrt[Gamma[1 + l - m]] Sqrt[Gamma[2 + l + m]]
        Sin[\[Theta]] SphericalHarmonicY[l, 1 + m, \[Theta], \[Phi]])/(
      Sqrt[Gamma[l - m]] Sqrt[Gamma[1 + l + m]]) - 
      l Sqrt[((1 + l - m) (1 + l + m))/(3 + 4 l (2 + l))]
        SphericalHarmonicY[1 + l, m, \[Theta], \[Phi]])

    In[51]:= fsf = %47;

    In[94]:= fsf /.  {l -> RandomInteger[73], m -> RandomInteger[99], 
\[Theta] -> RandomReal[{0, \[Pi]}], \[Phi] -> RandomReal[{0, 2 \[Pi]}]}

    Out[94]= (22.7394 + 0. I)/r

    In[48]:= FullSimplify[Coefficient[
      Sqrt[((l + m + 1) (l - m + 1))/((2 l + 3) (2 l + 
                1))] SphericalHarmonicY[l + 1, 
           m, \[Theta], \[Phi]] (f'[r] - l f[r]/r) + 
           Sqrt[((l + m) (l - m))/((2 l + 1) (2 l - 1))] (f'[
             r] + (l + 1) f[r]/r) - (Cos[\[Theta]] D[#, r] - 
           Sin[\[Theta]]/r D[#, \[Theta]]) &[f[r] SphericalHarmonicY[l, m, \[Theta], \[Phi]]], f'[r]], 
     Element[l | m, Integers]]

    Out[48]= Sqrt[((l - m) (l + m))/(-1 + 4 l^2)] - 
     Cos[\[Theta]] SphericalHarmonicY[l, m, \[Theta], \[Phi]] + 
     Sqrt[((1 + l - m) (1 + l + m))/(3 + 4 l (2 + l))]
       SphericalHarmonicY[1 + l, m, \[Theta], \[Phi]]

    In[54]:= fsfs = %48;

    In[85]:= fsfs /. {l -> RandomInteger[73], m -> RandomInteger[99], \[Theta] -> 
       RandomReal[{0, \[Pi]}], \[Phi] -> RandomReal[{0, 2 \[Pi]}]}

    Out[85]= 0.366093 - 0.00046913 I

An identity should turn the coefficients of the linear independent variables f[r], D[f[r],r] into zero.

The intention was to use the knowledge gained in verification of the relation to define an operator applicable in any order.

Thank you very much for the detailed answered. I am trying to digest all details and focus on the following piece which gives the correct answer when the operator 'gop' is applied twice

Clear[gop, singleQ]
singleQ[G_, f_] := 
 Length[Select[Level[f, {1}], Head[#] == G &]] == 1 /; 
  Head[f] === Times
singleQ[G_, f_] := 
 And @@ (singleQ[G, #] & /@ (List @@ f)) /; Head[f] === Plus
singleQ[G_, f_] := False /; Head[f] =!= Times && Head[f] =!= Plus
gop[G_, y_, f_] := 
 D[f /. G[l_, x__] -> l G[l + 1, x] + (l + 1) G[l - 1, x], 
   y] /; ! FreeQ[f, G] && FreeQ[Denominator[Together[f]], G] && 
   singleQ[G, Expand[f]] && FreeQ[D[f, y], Derivative[__][G]]
gop[G, y, gop[G, y, H[y] G[l, x]]]

Can you please tell what is the role of each one of the three lines above, which start with singleQ[G, f] := ...?

It doesn't take into account that the indices 'l' and 'm' have changed.

Right, right: A possible SphericalHarmonicY member in f must be detected by shRubinD and treated afterwards.

Possibly it's not (yet) correct because the second shRubinD does not take on the changed l and m indices of the SphericalHarmonicY from the previous derivative. Please check that.

Thanks for good intention, nevertheless I couldn't understand any of your answers.

Thanks. Can you please share more details? It is hard for me to follow.

Which pattern matching function ti use? How do I access the argument of Spherical Harmonic function which is multiplied by another function?

Thanks. Yet, the problem I like to solve seems to demand other solution.

As far as I see, the main difficulty is that the function needs to recognize the value of the first argument of the Spherical Harmonic function and then modify it accordingly. I don't know how to access the argument of the Spherical Harmonic function (or any other) in case it is multiplied by another function (which is relevant here according to the identity).

Thanks for the reply. Can you please elaborate what do you mean by define your function?