Interpolation and Commutant Lifting for Multipliers on Reproducing Kernel Hilbert Spaces

By Joseph A. Ball, T. Trent, Victor Vinnikov. 2001

211 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/694b4874b87ee4b286a9126bb48d2a23616084a5

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Citations in the corpus

Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis (2013, background): “the open unit disk is replaced by the polydisk, and [35–37] for further examples),”
Free holomorphic functions on the unit ball of B(H)n (2006, unknown): “) and its applications concerning interpolation in several variables ([26], [30], [32], [3], [15], [6], [36], etc.”
Conservative Structured Noncommutative Multidimensional Linear Systems (2005, result): “We mention [1, 2, 17, 15, 16] for extensions to the polydisk D n ⊂ C n setting, [33, 52, 10, 30, 3, 8, 18 , 45, 37, 4 ] for extensions to the unit ball B n ⊂ C n (where additional refinements concerning Nevanlinna-Pick-type interpolation and lifting theorems are also explored) ,a nd the recent work”
Some finite-dimensional backward-shift-invariant subspaces in the ball and a related interpolation problem (2002, background): “Operator-valued functions for which the kernel (1.2) is positive have been studied by Ball, Trent, and Vinnikov in [ 17 ], and some of our formulas are special cases of the general representation formulas obtained in [17], but the starting point and emphasis in [17] and in the present paper are diff”
Operator Theory on Noncommutative Domains (2007, unknown): “Let us mention that, recently, interpolation problems for F∞ n and related interpolation problems on the unit ball of C were also considered in [3], [14], [15], [16], [79], [80], [42], [82], etc.”
On the index of invariant subspaces in spaces of analytic functions of several complex variables (2005, background): “We refer the reader to [2], [5], [13], and [19].”
Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting (2011, methodology): “We will not pursue this line of idea here, which is also called the lurking isometry method; see [22], and postpone it to a work where we consider functions of several quaternionic variables.”
The corona theorem for the Drury–Arveson Hardy space and other holomorphic Besov–Sobolev spaces on the unit ball in $\C^n$ (2008, background): “ Cn that is a reproducing kernel Hilbert space with a complete irreducible Nevanlinna-Pick kernel (see [9] for the deﬁnition). The following Toeplitz corona theorem is due to Ball, Trent and Vinnikov [10] (see also Ambrozie and Timotin [2] and Theorem 8.57 in [9]). For f= (fα) N α=1 ∈ ⊕ NXand h∈ X, ”
Hilbert spaces of Dirichlet series and their multipliers (2003, background): “4 is a special case of the realization formula for general complete Pick kernels (see [4], [3], [18], [2]), and we omit the proof.”
Free holomorphic functions and interpolation (2007, background): “In the last two decades, significant progress has been made in noncommutative multivariable operator theory regarding noncommutative dilation theory and its applications to interpolation in several variables ([24], [28], [40], [38], [30], [4], [36]), and unitary invariants for n-tuples of operators ”
Complete Nevanlinna-Pick property of Dirichlet-type spaces (2002, background): “A general commutant lifting theorem for spaces with NP kernels is proved in [5].”
The Dirichlet space: a survey (2010, methodology): “The following Toeplitz Corona Theorem is due to Ball, Trent and Vinnikov [19] (see also Ambrozie and Timotin [5] and Theorem 8.”