The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $H$ be a complex Hilbert space. For $B\in\mathcal{L}(H)$ the powers are $B^{0}=I$ and $B^{k}=BB^{k-1}$ for $k\in\mathbb{N}$, and $S_{n}(B)=I+\sum_{k=1}^{n}B^{k}$ for $n\in\mathbb{N}$. For a bijection $T:H\to H$, $T^{-1}$ is its inverse map. Commutants and double commutants are those of The Commutant of a Set of Bounded Operators on a Complex Hilbert Space.

1. (Neumann series)¶ Let $B\in\mathcal{L}(H)$ with $\lVert B\rVert_{\mathrm{op}}<1$. Then $I-B$ is a bijection of $H$ onto $H$, $(I-B)^{-1}\in\mathcal{L}(H)$,

and, for every $n\in\mathbb{N}$,

in particular $S_{n}(B)\to(I-B)^{-1}$ in operator norm.

2. (Powers)¶ Let $X,Y\in\mathcal{L}(H)$ and let $c\ge0$ be real with $\lVert X\rVert_{\mathrm{op}}\le c$ and $\lVert Y\rVert_{\mathrm{op}}\le c$. Then $\lVert X^{k}\rVert_{\mathrm{op}}\le c^{k}$ and $\lVert X^{k}-Y^{k}\rVert_{\mathrm{op}}\le k\,c^{k-1}\lVert X-Y\rVert_{\mathrm{op}}$ for every $k\in\mathbb{N}$, where $c^{0}=1$.

3. (Inverses in double commutants)¶ Let $T\in\mathcal{L}(H)$ be a bijection of $H$ onto $H$ with $T^{-1}\in\mathcal{L}(H)$. Then $T^{-1}$ commutes with every $S\in\mathcal{L}(H)$ that commutes with $T$. Consequently, if $\mathcal{S}\subseteq\mathcal{L}(H)$ satisfies $\mathcal{S}=\mathcal{S}''$ and $T\in\mathcal{S}$, then $T^{-1}\in\mathcal{S}$.

4. (Invertibility of $A-iyI$)¶ Let $A\in\mathcal{L}(H)$ be self-adjoint and let $y\ne0$ be real. Then $\lVert(A-iyI)v\rVert^{2}=\lVert Av\rVert^{2}+y^{2}\lVert v\rVert^{2}$ for every $v\in H$; $A-iyI$ is a bijection of $H$ onto $H$; $(A-iyI)^{-1}\in\mathcal{L}(H)$; and $\lVert(A-iyI)^{-1}\rVert_{\mathrm{op}}\le|y|^{-1}$.