08/12/2013, 10:17 PM
Well I've been muddling this idea around for a while. I have been trying to create a hyper operator space and I recently realized the form of this. I'll start as follows:
If
is a hyper operator then, 
is a hyper operator created by forming left composition. I.e:
for all 
then
)
Associate to every function that is a finite product a number as follows:
)
Where
and p_n is the nth prime.
Now hyper operator space is the following:
} \in \mathbb{H})
Now define the inner product as follows:
 = \sum_{x=1}^\infty \sum_{y=1}^{\infty} f(x,y) \bar{g(x,y)})
Where quite clearly (f,f) converges for all elements since the terms decay to zero across x and y faster or just as fast as^2})
We say all the functions, 1/(2), 1/(3),...,\})
are dense in 
Orthonormalize them to get
such that:
 = \delta_{ij})
 \Delta_i)
Now we have the advantage of being in a Hilbert space and having an orthonormal basis.
The first operator we have is the transfer operator:
)
Since
this operator is well defined for any element of 
where  - 1][a_n])
Suppose:
exists such that 
for all values that [s] returns natural numbers at, this is our solution to hyper operators.
I think the key is to invesetigate the inner product.
If
Associate to every function that is a finite product a number as follows:
Where
Now hyper operator space is the following:
Now define the inner product as follows:
Where quite clearly (f,f) converges for all elements since the terms decay to zero across x and y faster or just as fast as
We say all the functions
Orthonormalize them to get
Now we have the advantage of being in a Hilbert space and having an orthonormal basis.
The first operator we have is the transfer operator:
Since
Suppose:
I think the key is to invesetigate the inner product.
