# Foundations of free noncommutative function theory A free holomorphic function on a free open set D is an nc-function that, in the free topology, is locally bounded. Free holomorphic functions are a class of nc-functions studied by the authors in [ 2 , 1 ]. There exists an auxiliary Hilbert space and an isometry. The following theorem gives sufficient conditions on a function to arise this way. Then the following are equivalent:. The series on the right-hand side of 17 that one gets by expanding the Neumann series of. This means each P k vanishes on a neighborhood of zero in every M n d , and hence must be zero. Repeat the argument of iii i of Theorem 1.

Then the proof of iii iv shows that F and F determine each other uniquely. So we get. John E. Europe PMC requires Javascript to function effectively.

## Foundations of Free Noncommutative Function Theory - fanifige.tk

Recent Activity. We characterize functions of d-tuples of bounded operators on a Hilbert space that are uniformly approximable by free polynomials on balanced open sets. The snippet could not be located in the article text. This may be because the snippet appears in a figure legend, contains special characters or spans different sections of the article. Eur J Math. Author manuscript; available in PMC Dec 1. PMID: Jim Agler and John E. McCarthy: ude. Copyright notice. Abstract We characterize functions of d -tuples of bounded operators on a Hilbert space that are uniformly approximable by free polynomials on balanced open sets.

Keywords: Nc functions, Free holomorphic functions, Intertwining preserving. Definition 3 Let be an infinite dimensional Hilbert space. The following are equivalent: The function F is intertwining preserving and sequentially strong operator continuous. Therefore it must be a scalar.

Proof By Proposition 1, if we restrict L to , we get an nc-function. Proof Any open ball centered at 0 is closed with respect to countable direct sums, so we can assume without loss of generality that D is closed with respect to countable direct sums and bounded.

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Proof Again we do this by induction on k. Definition 5 A free holomorphic function on a free open set D is an nc-function that, in the free topology, is locally bounded. Theorem 5 [ 1 , Theorem 8. Then the following are equivalent: The function F is intertwining preserving. Proof i iii This follows from Proposition 1. References 1. Global holomorphic functions in several noncommuting variables. Canad J Math. Pick interpolation for free holomorphic functions.

Amer J Math. The implicit function theorem and free algebraic sets. Trans Amer Math Soc. Matrix- J -unitary non-commutative rational formal power series. In: Alpay D, Gohberg I, editors. Operator Theory: Advances and Applications. Conservative structured noncommutative multidimensional linear systems.

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Dineen S. Springer Monographs in Mathematics. Springer; London: Complex Analysis on Infinite Dimensional Spaces. Analytic mappings between noncommutative pencil balls. J Math Anal Appl. Proper analytic free maps. J Funct Anal. Free analysis, convexity and LMI domains. Mathematical Methods in Systems, Optimization, and Control. Springer; Basel: Noncommutative ball maps. Every convex free basic semi-algebraic set has an LMI representation. Ann Math. Singularities of rational functions and minimal factorizations: the noncommutative and the commutative setting.

Linear Algebra Appl. Mathematical Surveys and Monographs. American Mathematical Society; Providence: Foundations of Free Noncommutative Function Theory. Free function theory through matrix invariants. Pascoe JE. The inverse function theorem and the Jacobian conjecture for free analysis.

Math Z. Free Pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables. Popescu G. Free holomorphic functions on the unit ball of B n.

## Noncommutative geometry

Free holomorphic functions and interpolation. Math Ann.

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Free holomorphic automorphisms of the unit ball of B n. J Reine Angew Math. Memoirs of the American Mathematical Society. Operator Theory on Noncommutative Domains. Taylor JL. Functions of several noncommuting variables. Bull Amer Math Soc. How does Europe PMC derive its citations network? Protein Interactions. Protein Families. Nucleotide Sequences.

The significance of free probability to random matrix theory lies in the fundamental observation that random matrices which are independent in the classical sense, also tend to be independent in the free probability sense, in the large limit. This is only possible because of the highly non-commutative nature of these matrices; as we shall see, it is not possible for non-trivial commuting independent random variables to be freely independent. Because of this, many tedious computations in random matrix theory, particularly those of an algebraic or enumerative combinatorial nature, can be done more quickly and systematically by using the framework of free probability, which by design is optimised for algebraic tasks rather than analytical ones.

For instance, if are freely independent and of expectation zero, then vanishes, but instead factors as. As a consequence, the behaviour of freely independent random variables can be quite different from the behaviour of their classically independent commuting counterparts. Nevertheless there is a remarkably strong analogy between the two types of independence, in that results which are true in the classically independent case often have an interesting analogue in the freely independent setting.

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For instance, the central limit theorem Notes 2 for averages of classically independent random variables, which roughly speaking asserts that such averages become gaussian in the large limit, has an analogue for averages of freely independent variables, the free central limit theorem , which roughly speaking asserts that such averages become semicircular in the large limit. Another important and closely related analogy is that while the distribution of sums of independent commutative random variables can be quickly computed via the characteristic function i.

This is strongly reminiscent of the appearance of the Stieltjes transform in random matrix theory, and indeed we will see many parallels between the use of the Stieltjes transform here and in Notes 4. As mentioned earlier, free probability is an excellent tool for computing various expressions of interest in random matrix theory, such as asymptotic values of normalised moments in the large limit.

Nevertheless, as it only covers the asymptotic regime in which is sent to infinity while holding all other parameters fixed, there are some aspects of random matrix theory to which the tools of free probability are not sufficient by themselves to resolve although it can be possible to combine free probability theory with other tools to then answer these questions. For instance, questions regarding the rate of convergence of normalised moments as are not directly answered by free probability, though if free probability is combined with tools such as concentration of measure Notes 1 then such rate information can often be recovered.

For similar reasons, free probability lets one understand the behaviour of moments as for fixed , but has more difficulty dealing with the situation in which is allowed to grow slowly in e. Because of this, free probability methods are effective at controlling the bulk of the spectrum of a random matrix, but have more difficulty with the edges of that spectrum as well as with related concepts such as the operator norm, Notes 3 as well as with fine-scale structure of the spectrum.

Finally, free probability methods are most effective when dealing with matrices that are Hermitian with bounded operator norm, largely because the spectral theory of bounded self-adjoint operators in the infinite-dimensional setting of the large limit is non-pathological. This is ultimately due to the stable nature of eigenvalues in the self-adjoint setting; see this previous blog post for discussion. For non-self-adjoint operators, free probability needs to be augmented with additional tools, most notably by bounds on least singular values, in order to recover the required stability for the various spectral data of random matrices to behave continuously with respect to the large limit.

We will discuss this latter point in a later set of notes. Blog at WordPress. Ben Eastaugh and Chris Sternal-Johnson. Subscribe to feed. What's new Updates on my research and expository papers, discussion of open problems, and other maths-related topics. Tag Archive. Then, one selects a -algebra of events modeled by subsets of , and assigns each of these events a probability in a countably additive manner, so that the entire sample space has probability. Finally, one builds commutative algebras of random variables such as complex-valued random variables, modeled by measurable functions from to , and assuming suitable integrability or moment conditions one can assign expectations to each such random variable. In the classical commutative setting, two bounded, real-valued random variables are independent if one has whenever are well-behaved functions such as polynomials such that , both vanish.

As a substitute, one can instead consider the notion of being freely independent or free for short , which means that whenever are well-behaved functions such that all of vanish. What is good mathematics? Why global regularity for Navier-Stokes is hard. Top Posts Almost all Collatz orbits attain almost bounded values Career advice Work hard Books Does one have to be a genius to do maths?

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## Foundations of Noncommutative Function Theory

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