The probelm I am trying to solve is the following:
Given: $X$, $Y$ are two Hilbert spaces. $T: X \to Y$ is a linear bounded operator, and the composite operator $TT^* :Y \to Y$ is invertible.
Question: Show that $x_1 := T^*(T T^*)^{-1}y$ is one solution of the operator equation $Tx = y$, for any $y \notin R(T)$
The way I have proceeded to solve this is by simply substituting $x_1$ into the equation $Tx = y$ and then having $TT^*$ cancel out. My question is what is the significance of $y \notin R(T)$? Would $y$ not have to be in the range in order for this to be a valid solution? Or have I solved it in an incorrect manner? Thank you