Proof of Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation

Each result cited is universally quantified over the data in its own statement. Norm bounds for sums, scalar multiples and composites 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, and adjoints are computed with Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, under which $\lVert T^{*}\rVert_{\mathrm{op}}=\lVert T\rVert_{\mathrm{op}}$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint. By The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §invertible, $\lVert R_{y}(A)\rVert_{\mathrm{op}}\le|y|^{-1}$ for all self-adjoint $A$ and real $y\ne0$. For $B\in\mathcal{L}(H)$ with $\lVert B\rVert_{\mathrm{op}}<1$, $S_{n}(B)$ is as in The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A. Since $t\mapsto t^{n+1}$ is nondecreasing and $t\mapsto(1-t)^{-1}$ is increasing on $[0,1)$, the bound of The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §neumann gives $\lVert(I-B)^{-1}-S_{n}(B)\rVert_{\mathrm{op}}\le\theta^{n+1}/(1-\theta)$ whenever $\lVert B\rVert_{\mathrm{op}}\le\theta<1$; this is how that clause is used below. Evaluation of polynomials is linear by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation and multiplicative by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism, so a linear combination of products of polynomials evaluates termwise.

Claim 1. By the adjoint calculus, $(A-iyI)^{*}=A^{*}+\overline{(-iy)}I=A+iyI$. If $T\in\mathcal{L}(H)$ is a bijection with $T^{-1}\in\mathcal{L}(H)$, then $(T^{-1})^{*}T^{*}=(TT^{-1})^{*}=I$ and $T^{*}(T^{-1})^{*}=(T^{-1}T)^{*}=I$, so $T^{*}$ is a bijection with inverse $(T^{-1})^{*}$. With $T=A-iyI$ this gives $R_{y}(A)^{*}=(A+iyI)^{-1}=R_{-y}(A)$. If $S\in\mathcal{L}(H)$ commutes with $A$, it commutes with $A-iyI$, hence with $R_{y}(A)$ by The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §commutant. Taking $S=A$ shows that $R_{y}(A)$ commutes with $A$; the same applied to $y'$ shows that $R_{y'}(A)$ commutes with $A$, hence with $R_{y}(A)$. If $\mathcal{S}=\mathcal{S}''$ and $A\in\mathcal{S}$, then $\mathcal{S}=(\mathcal{S}')'$ contains $I$ and is closed under sums and scalar multiples by Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §algebra, so $A-iyI\in\mathcal{S}$ and $R_{y}(A)\in\mathcal{S}$ by The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §commutant.

Claim 2. Write $R=R_{1}(a_{j})$. By the adjoint calculus, $\bigl(\tfrac12(R+R^{*})\bigr)^{*}=\tfrac12(R^{*}+R)$ and $\bigl(\tfrac{1}{2i}(R-R^{*})\bigr)^{*}=-\tfrac{1}{2i}(R^{*}-R)=\tfrac{1}{2i}(R-R^{*})$, so both entries are self-adjoint, and $\tfrac12(R+R^{*})+i\cdot\tfrac{1}{2i}(R-R^{*})=R$. Each entry has norm at most $\tfrac12(\lVert R\rVert_{\mathrm{op}}+\lVert R^{*}\rVert_{\mathrm{op}})=\lVert R\rVert_{\mathrm{op}}\le1$.

Claim 3. Since $R_{y}(A)(A-iyI)=I$ and $(B-iyI)R_{y}(B)=I$,

Claim 4. Let $y>0$ and write $R_{\pm}=R_{\pm y}(A)$. By Claim 1, $A$, $R_{+}$ and $R_{-}$ commute pairwise. Since $R_{-}(A+iyI)=I$ and $R_{+}(A-iyI)=I$,

where $A^{2}=AA$. Hence $\tfrac{y^{2}}{2}(R_{+}+R_{-})=y^{2}R_{+}R_{-}A$ and

Its norm is at most $\lVert A\rVert_{\mathrm{op}}^{3}\lVert R_{+}\rVert_{\mathrm{op}}\lVert R_{-}\rVert_{\mathrm{op}}\le\lVert A\rVert_{\mathrm{op}}^{3}y^{-2}$.

Claim 5. $\lVert(y'-y)R_{y}(A)\rVert_{\mathrm{op}}\le|y'-y|/y<1$, so by The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §neumann $I-B$, with $B=i(y'-y)R_{y}(A)$, is a bijection with inverse $G\in\mathcal{L}(H)$. Since $(A-iyI)R_{y}(A)=I$,

Then $(A-iy'I)\,GR_{y}(A)=(A-iyI)(I-B)GR_{y}(A)=I$ and $GR_{y}(A)(A-iy'I)=G(I-B)=I$, so $R_{y'}(A)=GR_{y}(A)$.

Claim 6. Fix $r>0$ and $\varepsilon>0$; put $y_{0}=r+1$ and $q=r/(r+1)<1$. Let $K$ be a complex Hilbert space and $C\in\mathcal{L}(K)$ self-adjoint with $\lVert C\rVert_{\mathrm{op}}\le r$. With $B_{0}=-iy_{0}^{-1}C$ we have $\lVert B_{0}\rVert_{\mathrm{op}}\le q$ and $C-iy_{0}I=-iy_{0}(I-B_{0})$, so $R_{y_{0}}(C)=iy_{0}^{-1}(I-B_{0})^{-1}$, and by The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §neumann

Let $u_{N}=iy_{0}^{-1}\bigl(1+\sum_{k=1}^{N}(-iy_{0}^{-1})^{k}x_{1}^{k}\bigr)\in\mathcal{P}_{1}$, where $x_{1}^{k}$ is the monomial of the word of length $k$ in the letter $1$; by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values and the linearity of evaluation, $u_{N}(C)=iy_{0}^{-1}S_{N}(B_{0})$. Next, $|1-y_{0}|=r<y_{0}$, so Claim 5 (with $y=y_{0}$, $y'=1$) gives $R_{1}(C)=(I-B_{1})^{-1}R_{y_{0}}(C)$ with $B_{1}=-irR_{y_{0}}(C)$, $\lVert B_{1}\rVert_{\mathrm{op}}\le r/y_{0}=q$, and by The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §neumann

using $S_{M}(B_{1})R_{y_{0}}(C)=\sum_{k=0}^{M}(-ir)^{k}R_{y_{0}}(C)^{k+1}$. Since $\lVert R_{y_{0}}(C)\rVert_{\mathrm{op}}\le1$ and $\lVert u_{N}(C)\rVert_{\mathrm{op}}\le1+q^{N+1}\le2$, The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §powers with $c=2$ gives $\lVert R_{y_{0}}(C)^{k+1}-u_{N}(C)^{k+1}\rVert_{\mathrm{op}}\le(k+1)2^{k}q^{N+1}$. Put $c_{M}=\sum_{k=0}^{M}r^{k}(k+1)2^{k}$, which depends only on $r$ and $M$. Choose $M$ with $q^{M+1}\le\varepsilon/2$ and then $N$ with $c_{M}q^{N+1}\le\varepsilon/2$ (possible by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric), and let $p=\sum_{k=0}^{M}(-ir)^{k}u_{N}^{k+1}\in\mathcal{P}_{1}$, with $u_{N}^{k+1}$ the $(k+1)$-fold product. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism, $p(C)=\sum_{k=0}^{M}(-ir)^{k}u_{N}(C)^{k+1}$, and therefore $\lVert R_{1}(C)-p(C)\rVert_{\mathrm{op}}\le q^{M+1}+c_{M}q^{N+1}\le\varepsilon$. The polynomial $p$ depends only on $r$ and $\varepsilon$.

Claim 7. For real $s\ge1$ let $\mathsf{A}(s)$ be the assertion: for every $\varepsilon>0$ there is $p\in\mathcal{P}_{2}$ with $\lVert R_{s}(C)-p(\mathbf{R}(C))\rVert_{\mathrm{op}}\le\varepsilon$ for every complex Hilbert space $K$ and every self-adjoint $C\in\mathcal{L}(K)$.

$\mathsf{A}(1)$ holds. By Claim 2, $R_{1}(C)=\mathbf{R}(C)_{1}+i\,\mathbf{R}(C)_{2}=p(\mathbf{R}(C))$ for $p=x_{1}+ix_{2}$, by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values.

If $1\le s\le s'\le\tfrac32s$ and $\mathsf{A}(s)$ holds, then $\mathsf{A}(s')$ holds. Fix $\varepsilon>0$ and put $h=s'-s$, so $0\le h\le s/2$. If $h=0$ there is nothing to prove. Otherwise, by Claim 5, $R_{s'}(C)=(I-B)^{-1}R_{s}(C)$ with $B=ihR_{s}(C)$, $\lVert B\rVert_{\mathrm{op}}\le h/s\le\tfrac12$, and The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §neumann gives

Put $c_{M}'=\sum_{k=0}^{M}h^{k}(k+1)2^{k}$. Choose $M$ with $2^{-M}\le\varepsilon/2$, then $\delta\in(0,1]$ with $c_{M}'\delta\le\varepsilon/2$, and by $\mathsf{A}(s)$ a polynomial $p\in\mathcal{P}_{2}$ with $\lVert R_{s}(C)-p(\mathbf{R}(C))\rVert_{\mathrm{op}}\le\delta$ for all $K$ and $C$. Then $\lVert p(\mathbf{R}(C))\rVert_{\mathrm{op}}\le1+\delta\le2$, and The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §powers gives $\lVert R_{s}(C)^{k+1}-p(\mathbf{R}(C))^{k+1}\rVert_{\mathrm{op}}\le(k+1)2^{k}\delta$. With $p'=\sum_{k=0}^{M}(ih)^{k}p^{k+1}\in\mathcal{P}_{2}$, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism gives $p'(\mathbf{R}(C))=\sum_{k=0}^{M}(ih)^{k}p(\mathbf{R}(C))^{k+1}$, so $\lVert R_{s'}(C)-p'(\mathbf{R}(C))\rVert_{\mathrm{op}}\le2^{-M}+c_{M}'\delta\le\varepsilon$ for all $K$ and $C$.

Conclusion. Given $s\ge1$, since $(2/3)^{l}\to0$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, there is $L\in\mathbb{N}$ with $(2/3)^{L}\le s^{-1}$, i.e. $(3/2)^{L}\ge s$. For $l\in\{0,1,\dots,L\}$ put $z_{l}=\min\{(3/2)^{l},s\}$, so $z_{0}=1$, $z_{L}=s$, and $z_{l}\le z_{l+1}\le\tfrac32z_{l}$ for $l<L$. By induction on $l$ (the inductive set being the set of $l\in\{0,\dots,L\}$ for which $\mathsf{A}(z_{l})$ holds), the two preceding paragraphs give $\mathsf{A}(z_{L})=\mathsf{A}(s)$.