[Paper Review] Noncommutative free universal monodromy, pluriharmonic conjugates, and plurisubharmonicity

J. E. Pascoe|arXiv (Cornell University)|Feb 18, 2020

Holomorphic and Operator Theory39 references4 citations

TL;DR

This paper establishes the Free Universal Monodromy theorem, proving that analytic continuation of noncommutative free functions on connected free sets is globally well-defined, which implies that pluriharmonic functions have global conjugates, locally invertible functions are globally invertible, and there is no nontrivial cohomology from analytic continuation. It generalizes the Dym-Helton-Klep-McCullough-Volcic theorem by showing that uniformly real analytic free functions are plurisubharmonic if and only if they decompose uniquely as the real part of an analytic function composed with a convex function.

ABSTRACT

We show that the monodromy theorem holds on arbitrary connected free sets for noncommutative free analytic functions. Applications are numerous-- pluriharmonic free functions have globally defined pluriharmonic conjugates, locally invertible functions are globally invertible, and there is no nontrivial cohomology theory arising from analytic continuation on connected free sets. We describe why the Baker-Campbell-Hausdorff formula has finite radius of convergence in terms of monodromy, and solve a related problem of Martin-Shamovich. We generalize the Dym-Helton-Klep-McCullough-Volcic theorem-- a uniformly real analytic free noncommutative function is plurisubharmonic if and only if it can be written as a composition of a convex function with an analytic function. The decomposition is essentially unique. The result is first established locally, and then Free Universal Monodromy implies the global result. Moreover, we see that plurisubharmonicity is a geometric property-- a real analytic free function plurisubharmonic on a neighborhood is plurisubharmonic on the whole domain. We give an analytic Greene-Liouville theorem, an entire free plurisubharmonic function is a sum of hereditary and antihereditary squares.

Motivation & Objective

• To establish the Free Universal Monodromy theorem for noncommutative free analytic functions on connected free sets.
• To prove that pluriharmonic free functions on connected domains admit globally defined pluriharmonic conjugates.
• To show that locally invertible free functions are globally invertible, eliminating nontrivial cohomology from analytic continuation.
• To generalize the Dym-Helton-Klep-McCullough-Volcic theorem by establishing a unique decomposition of uniformly real analytic free functions as the real part of a composition with a convex function.
• To demonstrate that plurisubharmonicity is a geometric property—plurisubharmonicity on a neighborhood implies plurisubharmonicity on the entire domain.

Proposed method

• The Free Universal Monodromy theorem is established by showing that analytic continuation along any path in a connected free set extends globally, leveraging the structure of free sets and uniform analyticity.
• The proof uses affine realizations of free functions via Hankel matrices and power series expansions to analyze the behavior of derivatives and conjugate derivatives.
• The decomposition of plurisubharmonic functions is constructed using a realization formula involving analytic and coanalytic functions, a contractive transfer function, and a positive semidefinite matrix kernel.
• The method relies on local representation via Lemma 3.3, which provides a canonical realization of a free plurisubharmonic function in terms of analytic, coanalytic, and contractive components.
• Observation 3.10 extends the realization to matrix amplifications, preserving the structure across levels, enabling global extension via monodromy.
• The uniqueness of the decomposition is proven by showing that any two such representations are equivalent up to unitary equivalence and junk terms, using canonical forms and invariance under unitary conjugation.

Experimental results

Research questions

• RQ1 Does the Free Universal Monodromy principle extend to algebraic contexts, as in free inverse function theorems?
• RQ2 Under what conditions does a nonsingular free noncommutative function admit a logarithm or a square root?
• RQ3 What is the theory of partial differential equations in free noncommutative function theory, particularly regarding existence and uniqueness?
• RQ4 Which Hankel-type matrices of power series coefficients exhibit geometric positivity, and is there a general theorem for such positivity?
• RQ5 Are all free plurisubharmonic functions necessarily real analytic, or can they be non-uniformly real analytic?

Key findings

• The Free Universal Monodromy theorem holds on arbitrary connected free sets, ensuring that analytic continuation of free functions is globally well-defined.
• Pluriharmonic free functions on connected domains admit globally defined pluriharmonic conjugates, eliminating local obstructions.
• Locally invertible free functions are globally invertible, and no nontrivial cohomology arises from analytic continuation on connected free sets.
• A uniformly real analytic free function is plurisubharmonic if and only if it decomposes uniquely as the real part of a composition of a convex function with an analytic function.
• Plurisubharmonicity is a geometric property: if a real analytic free function is plurisubharmonic on a neighborhood of a point, it is plurisubharmonic on the entire connected domain.
• An analytic Greene-Liouville theorem is established: every entire free plurisubharmonic function is a sum of hereditary and antihereditary squares.