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Hyper operator space - JmsNxn - 08/12/2013
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: 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 are dense in Orthonormalize them to get such that: 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 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. |