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

Each result cited is universally quantified over the data in its own statement. Norm bounds for sums, scalar multiples and composites, and $\lVert I\rVert_{\mathrm{op}}\le1$, are those of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations; that $\lVert T\rVert_{\mathrm{op}}$ is a bound for $T$ is Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound.

Claim 2 (Powers). We prove, by induction on $k$ (the inductive set being the set of $k\in\mathbb{N}$ for which both inequalities hold), that $\lVert X^{k}\rVert_{\mathrm{op}}\le c^{k}$ and $\lVert X^{k}-Y^{k}\rVert_{\mathrm{op}}\le kc^{k-1}\lVert X-Y\rVert_{\mathrm{op}}$. For $k=1$, $X^{1}=XI=X$ and $Y^{1}=Y$, so both hold. If they hold for $k$, then $\lVert X^{k+1}\rVert_{\mathrm{op}}=\lVert XX^{k}\rVert_{\mathrm{op}}\le c\cdot c^{k}$, and, since composition distributes over sums,

(with $\lVert Y^{k}\rVert_{\mathrm{op}}\le c^{k}$ by the first inequality applied to $Y$), so $\lVert X^{k+1}-Y^{k+1}\rVert_{\mathrm{op}}\le c\cdot kc^{k-1}\lVert X-Y\rVert_{\mathrm{op}}+\lVert X-Y\rVert_{\mathrm{op}}c^{k}=(k+1)c^{k}\lVert X-Y\rVert_{\mathrm{op}}$.

Claim 1 (Neumann series). Put $b=\lVert B\rVert_{\mathrm{op}}<1$. By Claim 2 with $X=B$ and $c=b$, $\lVert B^{k}\rVert_{\mathrm{op}}\le b^{k}$ for $k\in\mathbb{N}$, and $\lVert B^{0}\rVert_{\mathrm{op}}\le1$. For $n<n'$ in $\mathbb{N}$, $S_{n'}(B)-S_{n}(B)=\sum_{k=n+1}^{n'}B^{k}$, hence

the equality because $(1-b)\sum_{k=n+1}^{n'}b^{k}=b^{n+1}-b^{n'+1}$ (the sum telescopes). Since $b^{n}\to0$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, the sequence $(S_{n}(B))$ satisfies the Cauchy hypothesis of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §complete, which gives $T\in\mathcal{L}(H)$ with $S_{n}(B)\to T$ in operator norm. Letting $n'\to\infty$ in the displayed bound (the norm of the difference with $T$ is at most the norm of the difference with $S_{n'}(B)$ plus $\lVert S_{n'}(B)-T\rVert_{\mathrm{op}}$, which tends to $0$) gives $\lVert T-S_{n}(B)\rVert_{\mathrm{op}}\le b^{n+1}/(1-b)$ for every $n$, by Order Properties of Limits of Real Sequences.

Distributivity gives $(I-B)S_{n}(B)=S_{n}(B)(I-B)=I-B^{n+1}$ (the sums telescope). Hence

for every $n$, and the right side tends to $0$; a nonnegative real number bounded by every term of a sequence tending to $0$ is $0$ (Order Properties of Limits of Real Sequences). So $\lVert(I-B)T-I\rVert_{\mathrm{op}}=0$, i.e. $(I-B)T=I$ because the operator norm is a bound; likewise $T(I-B)=I$. Thus $I-B$ is a bijection of $H$ onto $H$ with inverse $T\in\mathcal{L}(H)$. Finally, telescoping as above, $\sum_{k=1}^{n}b^{k}=\frac{b-b^{n+1}}{1-b}\le\frac{b}{1-b}$, so $\lVert S_{n}(B)\rVert_{\mathrm{op}}\le1+\frac{b}{1-b}=\frac{1}{1-b}$, so $\lVert T\rVert_{\mathrm{op}}\le\frac{1}{1-b}+\frac{b^{n+1}}{1-b}$ for every $n$, and $\lVert T\rVert_{\mathrm{op}}\le\frac{1}{1-b}$ by Order Properties of Limits of Real Sequences.

Claim 3 (Inverses in double commutants). If $ST=TS$, then $T^{-1}S=T^{-1}S\,TT^{-1}=T^{-1}TS\,T^{-1}=ST^{-1}$. If $\mathcal{S}=\mathcal{S}''$ and $T\in\mathcal{S}$, every $S\in\mathcal{S}'$ commutes with $T$ by The Commutant of a Set of Bounded Operators on a Complex Hilbert Space §commutant, hence with $T^{-1}$; so $T^{-1}\in(\mathcal{S}')'=\mathcal{S}''=\mathcal{S}$.

Claim 4 (Invertibility of $A-iyI$). For $v\in H$, expanding the inner product (linear in the second and conjugate-linear in the first argument),

and $\langle v,Av\rangle=\langle Av,v\rangle$ because $A$ is self-adjoint; so the identity holds. In particular $\lVert(A-iyI)v\rVert\ge|y|\,\lVert v\rVert$ for every $v$, so $A-iyI$ is injective, and whenever $A-iyI$ is onto its inverse $G$ satisfies $\lVert Gw\rVert\le|y|^{-1}\lVert w\rVert$ for every $w$; $G$ is linear as the inverse of a linear bijection, so $G\in\mathcal{L}(H)$ with $\lVert G\rVert_{\mathrm{op}}\le|y|^{-1}$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound. The same holds with $y$ replaced by any real $y_{0}\ne0$.

It remains to prove surjectivity. Let $c=\lVert A\rVert_{\mathrm{op}}$, $t=1+(c+1)/|y|$ and $y_{0}=ty$; then $|y_{0}|=|y|+c+1>c$ and $|y-y_{0}|=c+1<|y_{0}|$. First, with $B_{0}=-iy_{0}^{-1}A$ we have $\lVert B_{0}\rVert_{\mathrm{op}}=c/|y_{0}|<1$ and $-iy_{0}(I-B_{0})=A-iy_{0}I$; by Claim 1, $I-B_{0}$ is a bijection of $H$, hence so is $A-iy_{0}I$, and by the previous paragraph its inverse $G_{0}$ lies in $\mathcal{L}(H)$ with $\lVert G_{0}\rVert_{\mathrm{op}}\le|y_{0}|^{-1}$. Second, with $B_{1}=i(y-y_{0})G_{0}$ we have $\lVert B_{1}\rVert_{\mathrm{op}}\le|y-y_{0}|/|y_{0}|<1$ and

By Claim 1, $I-B_{1}$ is a bijection of $H$, so $A-iyI$ is the composite of two bijections of $H$ and is onto. The first paragraph now gives $(A-iyI)^{-1}\in\mathcal{L}(H)$ and $\lVert(A-iyI)^{-1}\rVert_{\mathrm{op}}\le|y|^{-1}$.