DES : Is Double Encryption using same key twice equivalent to single DES using different key but of same size If we encrypt a plain text using Double DES but with same key K1,
Can we get the same result as if we would encrypt the plain text using single DES but with different key, say K2.
What I'm asking is can we achieve this for same size of K1 and K2 :
DES(K1,DES(K1,m)) = DES(K2,m)

What would be effective key bits for the first ?
I know this is the case for Mono-Alphabetic shift cipher but I'm confused for DES.
By Key Size I mean like 64 Bit key for which their are 2^55 computations.
Can we get the same result as if we would encrypt the plain text using single DES but with different key, say K2.

It has not been formally proven, but it is extremely plausible to assume that there isn't an equivalent 'double key', that is, for a K1, there isn't a K2 where DES(K1, DES(K1, M)) = DES(K2, M) for all values M (or even a number of values of M.
What has been proven (and is somewhat similar) is that there exist K1, K2 values for which there is no K3 for which DES( K1, DES( K2, M )) = DES( K3, M ) always holds; this is known as the 'DES is not a group' theorem (although it is typically expressed differently).  However, this result does not immediately imply what you are asking about.