As we know, the IV and the output of SHA-256 are identical in size. Suppose the input value to the SHA-256 is completely transparent. Is there any correlation between IV value and output? Is it possible to limit the output space?
No, you cannot limit the output space by altering the IV, if that's what you mean. The IV is a particular size, the inner state of the hash function which is identical to the native output size. You cannot just alter the inner state size without altering the hash function itself.
Fortunately, you don't need to. You can simply take as many bits from the output as you would like, and you'd still have the same inner strength of the hash function even though the size of the output does of course alter the strength of the resulting hash against pre-image and collision resistance.
As NIST states you would normally execute a specific formula to calculate the IV for a specific output size. This is specified in the NIST standard that defines SHA section 3.5.6 on SHA-512/t where $t$ is the output size.
The above doesn't clearly answer the additional question that is hiding in the middle:
Is there any correlation between IV value and output?
Yes, of course, any change to the initial value (IV) will directly affect all bits of the output. To restate the previous part of the answer: the size of the IV is fixed for SHA-256 and SHA-512 to the inner state size / native output size, 256 and 512 bits respectively.
