Invertible operator

Question

Let $K:V\to W$ such that $Kf = k$, where $V,W$ are infinite-dimensional Banach spaces. Is it correct to say that in general $f = (K^*K)^{-1}k$, however, when $V=W$, then $f = K^{-1}k$. $T^*$ here denotes the adjoint operator. In other words if $V\ne W$, we can't really talk about $K^{-1}$. I'm just trying to draw a parallel with $K$ being a real-valued matrix.

Answer 1

Inverse operators are a bit more abstract than what one may first think. The notation and ease of defining inverses from linear algebra is misleading. Really, the inverse operator (when it is well-defined) takes $w\in W$ and gives $v\in V$, with the added condition that $Kv = w$. This is the definition of the inverse operator. What the operator looks like really depends on the context. For example, the inverse operator to a differential operator is a Green's function which you can't nicely represent in terms of the PDE itself in the way you're suggesting. You could, in fact, define an inverse for a linear transformation from $V$ to $W$ if $\dim(W) < \dim(V)$ but you would need to make some specifications on the kinds of output you can have because the system is underdetermined. The way linear algebra is taught at the introductory level is actually a bit detrimental, I think. The notation and somewhat lack of clarity makes functional analysis a lot more difficult than it needs to be.