Not sure if this works in your context, but I would probably do something along these lines:

T[a_, b_, c_][f_[x_, y_]] := T[a, b, c*f[x + a, y + b]];
T[a_, b_] := T[a, b, 1]

Demonstration:

T[x, y][f[a, b]]
(* T[x, y, f[a + x, b + y]] *)

T[x, y][f[a, b]][g[c, d]][h[e, f]]
(* T[x, y, f[a + x, b + y] g[c + x, d + y] h[e + x, f + y]] *)

Alternatively, you could try this (so that T[x,y] is more explicit:

T[a_, b_, c_][f_[x_, y_]] := T[a, b, c*f[x + a, y + b]]);
T[a_, b_][f_[x_, y_]] := T[a, b, f[x + a, y + b]*T[a, b]];

T[1, 2][f[a, b]][g[c, d]]
(* T[1, 2, f[1 + a, 2 + b] g[1 + c, 2 + d] T[1, 2]] *)

And actually, there might be some advantages to the following:

T[a_, b_][prev_, f_[x_, y_]] := OperatorApplied[T[a, b], 2][prev*f[x + a, y + b]];
T[a_, b_][f_[x_, y_]] := T[a, b][1, f[x, y]];

T[x, y][f[a, b]][g[c, d]][h[e, f]]
(* OperatorApplied[T[x, y], 2][f[a + x, b + y] g[c + x, d + y] h[e + x, f + y]] *)

If you are willing to use Unprotect, you may try this way:

t[a_, b_][f_[x_, y_]] := f[x + a, y + b] t[a, b][#] &;
Unprotect[Function];
Function /: a_*Function[b_*t[args__][#]] := Function[a*b*t[args][#]];
t[x1, x2][f[a, b]]
t[x1, x2][f[a, b]][g[c, d]]
t[x1, x2][f[a, b]][g[c, d]][h[e, f]]

That worked for me, thank you very much. Best wishes.

That is right. The output should be an operator as well. Operator valued operations are used, for example, to calculate the commutator [x, del_x] (which is proportional to [x,p]) in quantum mechanics.

In your example, I would expect the output to be:

f[x+a] g[x+a] t[a]

Best wishes.

Thanks for the reply. However that would not work, because when "t[a,b]" acts of "f" there is no operator on the right hand side. What is in my mind is the following (for illustrative purposes for a single coordinate):

T[a] f[x] = f[x+a] T[a]

The original function I wrote kind of works, but I would like to remove the parantheses, e.g. in Out[25], and Flatten the expression. Hence f and g would be multiplied in the end, and there will be the T operator on the right hand side.