If T is a bounded and selfadjoint operator on a Hilbert spac

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.