Hello. Could you please help me to solve this problem:
is a linear unbounded operator on Hilbert space . If AU=UA for all - unitary operator on , then is bounded operator and .
Thus Ax is orthogonal to everything in the orthogonal complement of x, so that Ax must be a multiple of x. What's more, that multiple must be the same for x as it is for . Conclusion: A is a scalar multiple of the identity operator.