Invertability of quadratic function

Question

There is a proof in my lin alg book that shows that if T is a self adjoint operator, $T^2+Tb+c$ is invertible when $b^2 < 4c$?. I understand this proof.

Why is it that the above holds, and yet the function $f(x)=x^2+xb+c$ is not invertible? Can you not think of x as an identity operator, in which case it would be self adjoint?

Answer 1

Squaring a linear operator means composing it with itself. In particular, if an operator is the identity, then its square isn't the (nonlinear!) operator that squares something; its square is the identity operator.

Answer 2

$T^2$ means $T\circ T$ in the context of linear operators, while the $x^2$ means $x\cdot x$. In particular, $x\mapsto x^2+bx+c$ is not a linear map.