Suppose T is a linear transformation on V. Show that T is invertible iff there is a unique isometry S in L(V) such that T = S(root(T*T)). I'm assuming T is the adjoint transformation to T here. Thanks in advance for any help.
I'm assuming that V is a finite-dimensional vector space with an inner product $\displaystyle \langle.\,,\,.\rangle$, so that the adjoint of T is given by $\displaystyle \langle T^*x,y\rangle = \langle x,Ty\rangle\ (x,y\in V)$. The operator $\displaystyle R = \sqrt{T^*T}$ is selfadjoint, and is invertible if and only if T is (reason: take determinants in the equation $\displaystyle R^2=T^*T$). Also, for each $\displaystyle x\in V$,
$\displaystyle \|Rx\|^2 = \langle Rx,Rx\rangle = \langle R^2x,x\rangle = \langle T^*Tx,x\rangle = \langle Tx,Tx\rangle = \|Tx\|^2.\qquad(*)$
So there is a (unique) isometric linear map $\displaystyle S_0$ from the range of R to the range of T defined by $\displaystyle S_0(Rx) = Tx$. If T and R are invertible then these ranges are the whole of V, and $\displaystyle S_0$ is the unique isometry on V such that $\displaystyle T=S_0R$.
Now suppose that T is not invertible. Then $\displaystyle S_0$ is still an isometry, but neither its domain nor its range are the whole of V. However, since the ranges of T and R have the same dimension, so do their orthogonal complements. Therefore there exists a (nonzero) isometry $\displaystyle S_1$ from $\displaystyle (R(V))^\perp$ to $\displaystyle (T(V))^\perp$. The operators $\displaystyle S_+ = S_0+S_1$ and $\displaystyle S_- = S_0-S_1$ are then distinct isometries on the whole of V, and they both have the property that $\displaystyle S_\pm R = T$. So in this case, S is not unique.