2016
DOI: 10.1016/j.jfa.2016.06.010
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Noncommutative reproducing kernel Hilbert spaces

Abstract: Abstract. The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and operator theory. An interesting generalization of holomorphic functions, namely free noncommutative functions (e.g., functions of square-matrix arguments of arbitrary size satisfying additional natural compatibility conditions), is now an active area of research, with mo… Show more

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Cited by 51 publications
(72 citation statements)
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References 44 publications
(84 reference statements)
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“…If one replaces the commutative free Nullstellensatz with the noncommutative homogeneous Nullstellensatz, then the same argument as above (where polynomials are replaced by free polynomials) gives the following theorem: 6 Theorem 2.2. Let I and J be two homogeneous ideals in C z .…”
Section: A Digression -The Purely Algebraic Casementioning
confidence: 99%
“…If one replaces the commutative free Nullstellensatz with the noncommutative homogeneous Nullstellensatz, then the same argument as above (where polynomials are replaced by free polynomials) gives the following theorem: 6 Theorem 2.2. Let I and J be two homogeneous ideals in C z .…”
Section: A Digression -The Purely Algebraic Casementioning
confidence: 99%
“…One can modify the argument to get a realization formula for B(K, L)-valued bounded nc-functions on B δ , or to prove Leech theorems (also called Toeplitzcorona theorems-see [10] and [8]). For finite-dimensional K and L, this was done in [2]; for infinite-dimensional K and L the formula was proved in [6], using results from [7].…”
Section: Closing Remarksmentioning
confidence: 99%
“…is then an operator-valued free formal kernel in the sense of [4,3] (see also [8] which develops free Aleksandrov-Clark theory using the free formal RKHS setup). If F (Z) := α FαZ α ∈ Hnc(K), then for any α ∈ F d , the linear H−valued map defined by coefficient evaluation:…”
Section: Coefficient Evaluation and Free Formal Rkhs Let K Be An Opementioning
confidence: 99%