Operatornorm of self-adjoint operator

Question

Let $A$ be a self-adjoint, positive definite operator in $H$ with $\inf\, \text{spec}(A)\geq 1$. I want to show that it is equivalent to the conditions that $A$ is invertible and $||A^{-1}||\leq1$. Can someone help?

Answer 1

If $|\lambda| <1$ then $\|\lambda A^{-1}\| <1$. For any operator $S$ with $\|S\| <1$ the operator $(I-S)$ is invertible (the inverse being $\sum_{n=0}^{\infty} S^{n}$). Hence $I-\lambda A^{-1}$ is invertible which implies that $A-\lambda I$ is invertible. Hence $\lambda$ is not in the spectrum of $A$.

This proves that $|\lambda| \geq 1$ for every $\lambda$ in the spectrum. The spectrum of a positive operator is contained in $[0,\infty)$. Hence $\lambda \geq 1$.