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Canadian journal of mathematics, ISSN 0008-414X, 04/2015, Volume 67, Issue 2, pp. 241 - 285
We define a free holomorphic function to be a function that is locally, with respect to the free topology, a bounded nc-function. We prove that free... 
Award Winners | Noncommutative analysis | Free holomorphic functions | INTERPOLATION | MATHEMATICS | free holomorphic functions | noncommutative analysis | DOMAINS | UNIT BALL
Journal Article
The Journal of geometric analysis, ISSN 1559-002X, 2018, Volume 29, Issue 4, pp. 3039 - 3054
... Jim Agler 1 · Zinaida Lykova 2 · N. J. Young 2,3 Received: 21 February 2018 / Published online: 21 August 2018 © The Author(s) 2018 Abstract In this paper... 
Carathéodory extremal problem | 53C22 | 30E05 | Mathematics | Bidisc | Primary: 32A07 | 54C15 | Complex geodesics | 32F45 | Abstract Harmonic Analysis | Symmetrized | Fourier Analysis | Secondary: 47A25 | Convex and Discrete Geometry | 47A57 | Global Analysis and Analysis on Manifolds | Kobayashi extremal problem | Differential Geometry | Dynamical Systems and Ergodic Theory
Journal Article
Transactions of the American Mathematical Society, ISSN 0002-9947, 05/2016, Volume 368, Issue 5, pp. 3157 - 3175
We prove an implicit function theorem for non-commutative functions. We use this to show that if p(X, Y) is a generic non-commuting polynomial in two variables... 
NC functions | Free algebraic sets | Free holomorphic functions | MATHEMATICS | free holomorphic functions | free algebraic sets | FREE HOLOMORPHIC-FUNCTIONS | UNIT BALL
Journal Article
American journal of mathematics, ISSN 1080-6377, 2015, Volume 137, Issue 6, pp. 1685 - 1701
We give necessary and sufficient conditions to solve an interpolation problem for free holomorphic functions bounded in norm on a free polynomial polyhedron.... 
Interpolation | Holomorphic functions | Functions of complex variables | POLYNOMIALS | MATHEMATICS | EXTENSION | UNIT BALL | SUMS | Functions | Polynomials | Functional equations | Analysis | Mathematical functions | Mathematics | Polyhedra
Journal Article
Advances in mathematics (New York. 1965), ISSN 0001-8708, 2018, Volume 328, pp. 133 - 159
The set G=def{(z+w,zw):|z|<1,|w|<1}⊂C has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a... 
Distinguished boundary | Convexity | Symmetrized bidisc | Extreme point | Inner function
Journal Article
Transactions of the American Mathematical Society, ISSN 0002-9947, 03/2014, Volume 366, Issue 3, pp. 1379 - 1411
can occur. We give an extension of Hamburger's results to Pick functions of two variables.]]> 
Integers | Infinity | Scalars | Hilbert spaces | Mathematical vectors | Polynomials | Mathematical moments | Matrices | Mathematical functions | Degrees of polynomials | INTERPOLATION | BIDISK | MATHEMATICS
Journal Article
2002, Graduate studies in mathematics, ISBN 9780821828984, Volume 44, xix, 308
Book
Journal of mathematical analysis and applications, ISSN 0022-247X, 2017, Volume 453, Issue 1, pp. 227 - 240
We prove a realization formula and a model formula for analytic functions with modulus bounded by 1 on the symmetrized bidiscG=def{(z+w,zw):|z|<1,|w|<1}. As an... 
Hilbert space model | Pick theorem | Analytic functions | Schur class | INTERPOLATION | MATHEMATICS | MATHEMATICS, APPLIED
Journal Article
Annals of mathematics, ISSN 0003-486X, 2012, Volume 176, Issue 3, pp. 1783 - 1826
We prove generalizations of Löwner's results on matrix monotone functions to several variables. We give a characterization of when a function of d variables is... 
Mathematical monotonicity | Analytic functions | Eigenvalues | Hilbert spaces | Mathematical functions | Mathematical vectors | Eigenvectors | Commuting | Monotonic functions | MATHEMATICS | BOUNDARY | NEVANLINNA-PICK INTERPOLATION | THEOREM
Journal Article
Journal of mathematical analysis and applications, ISSN 0022-247X, 2019, Volume 473, Issue 2, pp. 1377 - 1413
The symmetrized bidiscG=def{(z+w,zw):|z|<1,|w|<1} has interesting geometric properties. While it has a plentiful supply of complex geodesics and of... 
Geodesic | Automorphism | Complex manifold | Cohomogeneity one | MATHEMATICS | MATHEMATICS, APPLIED | COMPLEX GEODESICS | METRICS | AUTOMORPHISMS | Mathematics - Complex Variables
Journal Article
Journal of functional analysis, ISSN 0022-1236, 2016, Volume 270, Issue 8, pp. 3000 - 3046
We generalize to several variables the classical theorem of Nevanlinna that characterizes the Cauchy transforms of positive measures on the real line. We show... 
Cauchy transform | Self-adjoint operator | Resolvent | Pick class | COMMUTING CONTRACTIONS | MATHEMATICS | REALIZATIONS | DILATIONS
Journal Article
MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY, ISSN 0065-9266, 03/2019, Volume 258, Issue 1242, pp. 1 - 1
A set V in a domain U in C-n has the norm-preserving extension property if every bounded holomorphic function on V has a holomorphic extension to U with the... 
SUBVARIETIES | spectral set | COMPLEX GEODESICS | AUTOMORPHISMS | norm-preserving extension property | semialgebraic set | INTERPOLATION | MATHEMATICS | Caratheodory extremal problem | von Neumann inequality | complex geodesic | HOLOMORPHIC-FUNCTIONS | Symmetrized bidisc | Kobayashi extremal problem | holomorphic retract | GEOMETRY
Journal Article
Canadian mathematical bulletin, ISSN 0008-4395, 09/2018, Volume 61, Issue 3, pp. 458 - 463
We give a new proof that bounded non-commutative functions on polynomial polyhedra can be represented by a realization formula, a generalization of the... 
MATHEMATICS | non-commutative function | realization formula | Mathematics - Functional Analysis
Journal Article
Annals of mathematics, ISSN 0003-486X, 7/2014, Volume 180, Issue 1, pp. 403 - 405
We fix a gap in the proof of Theorem 7.24 in Ann. of Math. 176 (2012), 1783–1826. 
Mathematical variables | Mathematical functions | Mathematical theorems | Monotonic functions
Journal Article
Concrete Operators, ISSN 2299-3282, 04/2016, Volume 3, Issue 1, pp. 15 - 24
We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable. 
Free holomorphic function | Noncommutative function
Journal Article
Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 03/2017, Volume 447, Issue 2, pp. 1163 - 1196
LetΓ=def{(z+w,zw):|z|≤1,|w|≤1}⊂C2. A Γ-inner function is a holomorphic map h from the unit disc D to Γ whose boundary values at almost all points of the unit... 
Pick matrix | Blaschke product | Interpolation | Complex geodesic | Symmetrized bidisc | MATHEMATICS, APPLIED | NEVANLINNA-PICK INTERPOLATION | THEOREM | COMPLEX GEODESICS | POLYDISK | BOUNDARY INTERPOLATION | AUTOMORPHISMS | CONTRACTIONS | MATHEMATICS | DIMENSIONS | EXTREMAL HOLOMORPHIC MAPS | Mathematics - Complex Variables
Journal Article
Acta Mathematica, ISSN 0001-5962, 09/2010, Volume 205, Issue 1, pp. 1 - 3
...  Acta Math., 205 (2010), 13 DOI: 10.1007/s11511-010-0049-4 c [circlecopyrt] 2010 by Institut Mittag-Le er. All rights reserved Jim Agler University... 
Journal Article
Acta mathematica, ISSN 0001-5962, 2005, Volume 194, Issue 2, pp. 133 - 153
...Acta Math., 194 (2005), 133-153 Distinguished varieties by JIM AGLER and JOHN E. MCCARTHY University of California, San Diego Washington University St. Louis... 
Mathematics, general | Mathematics | BIDISK | MATHEMATICS | NEVANLINNA-PICK INTERPOLATION | SPACES
Journal Article
Proceedings of the London Mathematical Society, ISSN 0024-6115, 2013, Volume 106, Issue 4, pp. 781 - 818
We introduce the class of n‐extremal holomorphic maps, a class that generalizes both finite Blaschke products and complex geodesics, and apply the notion to... 
INTERPOLATION | MATHEMATICS | COMPLEX GEODESICS | COMMUTANT LIFTING THEOREM | NEVANLINNA-PICK PROBLEM | POLYDISK | GEOMETRY | Discs | Interpolation | Maps | Analytic functions | Mathematical analysis | Classification | Disks | Boundaries
Journal Article
Journal of Operator Theory, ISSN 0379-4024, 1/2012, Volume 67, Issue 1, pp. 215 - 236
We consider pairs of commuting isometries that are annihilated by a polynomial. We show that the polynomial must be inner toral, which is a geometric condition... 
Riemann surfaces | Algebra | Mathematical theorems | Analytic functions | Hilbert spaces | Coordinate systems | Polynomials | Mathematical functions | Commuting | MATHEMATICS | SPACES | SUBNORMAL-OPERATORS | DISTINGUISHED VARIETIES
Journal Article
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