If T is a bounded and self-adjoint operator on a Hilbert space and T_2 = T, show that T is the orthogonal projection onto its range. Solution Since T is bounded and also self joint we have T = T* and T* = T Since TT* = T*T we get TT = T*T* i.e. T is normal. So it follows that T is the orthogonal projection onto its range..

1. If T is a bounded and self-adjoint operator on a Hilbert space and T_2 = T, show that T is the
orthogonal projection onto its range.
Solution
Since T is bounded and also self joint we have
T = T* and T* = T
Since TT* = T*T
we get TT = T*T*
i.e. T is normal.
So it follows that T is the orthogonal projection onto its range.