Exercise 6.6.1

Answers

1.: No. Orthogonal projection is self-adjoint by Theorem 6.24. But for general projection the statement is not true. For example, the transformation $T (a, b) = (a + b, 0)$ is a projection which is not self-adjoint.
2.: Yes. See the paragraph after Definition of “orthogonal projection”.
3.: Yes. This is the result of the Spectral Theorem.
4.: No. It’s true for orthogonal projection but false for general projection. For example, the the transformation $T (a, b) = (a + b, 0)$ is a projection on $W$ . But we have $T (0, 1) = (1, 0)$ is not the point closest to $(0, 1)$ since $(0, 0)$ is much closer.
5.: No. An unitary operator is usually invertible. But an projection is generally not invertible. For example, the mapping $T (a, b) = (a, 0)$ .

jephianlin

2011-06-27 00:00