Unitary transformations

The three most obvious pleasant relations that a linear transformation on an inner product space can have to its adjoint are that they are equal (Hermitian), or that one is the negative of the other (skew), or that one is the inverse of the other (not yet discussed). The word that describes the last of these possibilities is unitary: that's what a linear transformation U is called in case it is invertible and U−1 = U*. The definition can be expressed in a “less prejudiced” way as U * U = 1—less prejudiced in the sense that it assumes less—but it is not clear that the less prejudiced way yields just as much. Does it?

Problem 139. If U is a linear transformation such that U * U = 1, does it follow that U * U = 1?

Unitary matrices

It seems fair to apply the word “unitary” to a matrix in case the linear transformation it defines is a unitary one. (Caution: when language that makes sense in inner product spaces only is applied to matrices, the basis that establishes the correspondence between matrices and linear transformations had better be an orthonormal one.) A quick glance usually suffices to tell whether or not a matrix is Hermitian; is there a way to tell by looking at a matrix whether or not it is unitary? The following special cases are a fair test of any proposed answer to the general question.