The author’s proof of the Bieberbach conjecture [Acta Math. 154, 137–152 (1985; Zbl 0573.30014)] was originally conceived in a functional analysis context. The functional analysis view is a fascinating one which opens new directions for research in canonical models, linear systems, and interpolatory function theory. In the present paper, the author explains and expands on the original conception.
The conception is based on the canonical models approach to invariant subspace theory by the author and the reviewer [Square summable power series. New York etc.: Holt, Rinehart and Winston (1966; Zbl 0153.39602); Perturb. Theory Appl. Quantum Mech., Proc. Adv. Sem. Math. Res. Center US Army, Univ. Wisconsin, Madison 1965, 295–392 (1966; Zbl 0203.45101)]. Let \(\mathcal C(z)\) be the space of square summable power series with complex coefficients, and let \(B(z)\) be a power series such that multiplication by \(B(z)\) is a contractive transformation of \(\mathcal C(z)\) into itself. Let \(\mathcal M(B)\) be the range of multiplication by \(B(z)\) in the scalar product which makes the transformation isometric. The complementary space \(\mathcal H(B)\) is the set of elements \(f(z)\) of \(\mathcal C(z)\) such that \[ \|f(z)\|^2_{\mathcal H(B)}=\sup\Big[\|f(z)+B(z)g(z)\|^2_{\mathcal{C}(z)}-\|g(z)\|^2 _{\mathcal{C}(z)}\Big] \] is finite, where the supremum is over all \(g(z)\) in \(\mathcal C(z)\). A linear system \(A,B,C,D\) is obtained with state space \({\mathcal H}(B)\) and external space \(\mathcal C\) (in this case, the complex numbers). The system has main transformation \(A\colon f(z)\mapsto[f(z)-f(0)]/z\) in \(\mathcal H(B)\), output transformation \(B\colon f(z)\mapsto f(0)\) from \(\mathcal H(B)\) to \(\mathcal C\), input transformation \(C\colon c\mapsto[B(z)-B(0)]c/z\) from \(\mathcal C\) to \(\mathcal H(B)\), and external transformation \(D\colon c\mapsto B(0)c\) on \(\mathcal C\). The system is conjugate isometric in the sense that the matrix \(\left[ \begin{smallmatrix} A &C\\ C &D \end{smallmatrix} \right]\) has an isometric adjoint.
A unitary system \(A,B,C,D\), that is, one for which \(\left[ \begin{smallmatrix} A &B\\ C &D \end{smallmatrix} \right]\) is a unitary matrix, is constructed in a similar way from an extension space \(\mathcal D(B)\) for \(\mathcal H(B)\). Unitary systems are equivalent to the model of B. Sz.-Nagy and C. Foiaş [“Harmonic analysis of operators on Hilbert space” (North-Holland, Amsterdam) (1970; Zbl 0201.45003)]. Recent characterizations of unitary systems among conjugate isometric systems have been given by J. A. Ball and T. L. Kriete [Integral Equations Oper. Theory 10, 17–61 (1987; Zbl 0621.47012)], N. K. Nikol’skiĭ and V. I. Vasyunin [“Notes on two function models”, in: The Bieberbach conjecture (West Lafayette, 1985), Math. Surv. Monogr. 21, 113–141 (1986; MR 88f:47008)], and D. Sarason [ibid., 153–166 (1986; MR 88d:47014a)].
A normalized Riemann mapping function is an analytic function on the unit disk which takes distinct values at distinct points and has value zero and positive derivative at the origin. For any real number \(\nu\), let \(\mathcal G^\nu\) be the Kreĭn space of generalized power series \(f(z)=\sum_{n=1}^\infty a_nz^{\nu+n}\) with finite scalar self-product \(\sum_{n=1}^\infty(\nu+n)|a_n|^2\). A space \(\mathcal G_r^\nu\), \(r\) any positive integer, is defined by identifying series in \(\mathcal G^\nu\) whose coefficients coincide for all indices \(n\) not greater than \(r\). If \(B(z)\) is a normalized Riemann mapping function for a subregion of the unit disk, then substitution by \(B(z)\) is a contractive transformation on \(\mathcal G^\nu\) for every \(\nu\). By a truncation of inequalities, the transformation also acts contractively in \(\mathcal G_r^\nu\) if \(r+1+\nu\geq0\).
Let \(B(z)\) be a normalized Riemann mapping function for a subregion of the unit disk, and consider the case of the Dirichlet space \(\mathcal G=\mathcal G^0\). The set of series \(f(B(z))\) with \(f(z)\) in \(\mathcal G\) is a Hilbert space \(\mathcal M(B)\) in the scalar product such that substitution by \(B(z)\) is an isometry from \(\mathcal G\) onto \(\mathcal M(B)\). A complementary space \(\mathcal G(B)\) is defined as before. The theory of spaces \(\mathcal G(B)\) is a logarithmic counterpart to the theory of spaces \(\mathcal H(B)\). For example, the reproducing kernel for a space \(\mathcal H(B)\) is \([1-B(z)\overline{B}(w)]/(1-z\overline{w})\). The reproducing kernel for \(\mathcal G(B)\) is \(\log[1-B(z)\overline{B}(w)]/(1-z\overline{w})\). Another example is the contractive transformation \(f(z)\) into \(\tilde{f}(z)\) from \(\mathcal H(B)\) to \(\mathcal H(B^*)\); \(B^*(z)=\overline{B}(\overline{z})\). Its action on kernel functions is \([1-B(z)\overline{B}(w)]/(1-z\overline{w})\) into \([B^*(z)-\overline{B}(w)]/(z-\overline{w})\). The logarithmic counterpart of the tilde transformation is the contractive Grunsky transformation \(U\) from \(\mathcal G(B)\) to \(\mathcal G(B^*)\). Its action on kernel functions is \(\log[1-B(z)\overline{B}(w)]/(1-z\overline{w})\) into \(\log[B'(0)/B^*(z)-B'(0)/\overline{B}(w)]/(1/z-1/\overline{w})\).
The spaces \(\mathcal G(B)\) are thus parallel to the state spaces for conjugate isometric systems. Extension spaces \(\mathcal I(B)\) exist which play a similar role with respect to unitary systems.
A logarithmic counterpart to the Carathéodory-Fejér interpolation problem is formulated. Let \(B(z)=B_1z+\cdots+B_rz^r\) be a polynomial with constant term zero and coefficient of \(z\) positive. It is conjectured that \(B(z)\) has an extension to a normalized Riemann mapping function for a subregion of the unit disk if substitution by \(B(z)\) is contractive in \(\mathcal G_r^\nu\) for every nonpositive integer \(\nu\) such that \(r+1+\nu\geq0\).
The paper contains no formal theorems or proofs. Readers who are familiar with canonical models or are willing to take some things on faith can easily follow the main lines of development. Full details have yet to appear in published form.