R7 Holomorphic operator substitution
On this page · Permalink · Source line 5903
Reader roadmap
The scalar function, zero-free domain, and logarithm branch are fixed before substituting an operator. The bounded Banach-space functional calculus here is separate from the finite canonical-jet contract of Part I. Keep those spaces and hypotheses visible when using the reciprocal family.
This section receives the operator-substitution result deliberately moved out of Paper IV. Fix an integer \(N\geq1\) and the entire scalar function
More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain \(U\), choose \(g=\log{\mathsf D}_N\) once and define \({\mathsf S}_N^{(r)}(z)=\exp(-r g(z))\) for complex \(r\). The bounded Banach-space result below uses classical holomorphic functional calculus; it is separate from the finite canonical-jet calculus formalized in Part I.
Standard fact R7.1: Holomorphic functional calculus
Let \(A\) be a bounded operator on a complex Banach space. If \(f\) is holomorphic on a neighborhood of \({\operatorname{Spec}}(A)\), then
for an admissible positively oriented contour \(\Gamma\) lying in the holomorphy domain and the resolvent set, with winding number one about the spectrum and zero outside the domain. The calculus preserves sums, products, and composition. We use the standard form from [DunfordSchwartz1958]; an accessible proof is given in [TaylorBanachAlgebras], Appendix A.
Permalink · Source line 5915 · Claim record · Lean coverage · Code · Worked examples
Proposition R7.2: Operator reciprocal family
Let \(A\) be bounded and suppose \({\operatorname{Spec}}(A)\) lies in a simply connected region on which \({\mathsf D}_N\) has no zeros. Fix a holomorphic branch \(g=\log{\mathsf D}_N\) there. Then
is well defined and
If \((P_i)\) is a finite complete family of bounded idempotents, then
for every holomorphic \(f\) on a neighborhood of the spectrum.
Read proof · Read statement and proof side by side · Permalink · Source line 5925 · Claim record · Lean coverage · Code · Worked examples
Proof of Proposition R7.2
The spectral mapping theorem makes \({\mathsf D}_N(A)\) invertible. The product identity follows from multiplicativity of the functional calculus and the fixed logarithm. Applying \(P_i\) and \(P_j\) to the Cauchy integral formula gives the packet formula.
Back to statement · Read statement and proof side by side · Permalink · Source line 5951 · Claim record · Lean coverage · Code · Worked examples