Free analysis questions II: The Grassmannian completion and the series expansions at the origin

By D. Voiculescu. 2008

128 citations recorded by the source corpus.

Abstract The fully matricial generalization in part I, of the difference quotient derivation on holomorphic functions, in which ℂ is replaced by a Banach algebra B, is extended from the affine case to a Grassmannian completion. The infinitesimal bialgebra duality, the duality transform generalizing the Stieltjes transform and the spectral theory with non-commuting scalars all extend to this completion. The series expansions of fully matricial analytic functions are characterized, providing a new way to generate fully matricial functions.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f43aff5679e12dfcf63606417d294f09f031c10d

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Citations in the corpus

Free Probability and Random Matrices (2014, background): “One should, however, note that the analytic theory of operator-valued free convolution lacks at the moment some of the deeper statements of the scalar-valued theory; developing a reasonable analogue of complex function theory on an operatorvalued level, addressed as free analysis, is an active area ”
Noncommutative reproducing kernel Hilbert spaces (2016, background): “ Taylor [47] (see also [48]) as a setting for the study of a functional calculus for tuples of freely noncommuting matrix variables, and independently in the work of Voiculescu and collaborators (see [49, 50]) in the context of the needs of a free-probability theory, as well as in the work of Helton”
Pick Interpolation for free holomorphic functions (2013, background): “ational Science Foundation Grant DMS DMS 1300280 1 Nc-functions have been studied for a variety of reasons: by Taylor [25], in thecontext ofthefunctionalcalculus fornon-commutingoperators; Voiculescu [26, 27], in the context of free probability; Popescu [19, 20, 21, 22], in the context of extending ”
Interpolation and transfer-function realization for the noncommutative Schur-Agler class (2016, background): “Closely related approaches to and results on such a “free noncommutative function theory” can be found in the work of Voiculescu [81, 82] and Helton-Klep-McCullough [45, 46, 47].”
Global Holomorphic Functions in Several Noncommuting Variables (2015, unknown): “This has been done in a variety of ways: by Taylor [23], in the context of the functional calculus for non-commuting operators; Voiculescu [24, 25], in the context of free probability; Popescu [18, 19, 20, 21], in the context of extending classical function theory to d-tuples Partially supported by ”
The inverse function theorem and the Jacobian conjecture for free analysis (2013, background): “s in [7]. They obtained the formula (2.1) f X H 0 X = f(X) Df(X)[H] 0 f(X) . These types of formulas are pervasive throughout the free analysis literature. For other references see Voiculescu [14] and the recently completed tome by D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov [9]. Formula 2.1 c”