On the Bessmertnyĭ Class of Homogeneous Positive Holomorphic Functions of Several Variables

Kalyuzhnyĭ-Verbovetzkiĭ, Dmitry S.

doi:10.1007/978-3-0348-7881-4_12

Cited by 11 publications

(35 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our work in this direction should be compared to that of Ambrozie and Timotin [5], Ball and Bolotnikov [9] and Kalyuzhnyi-Verbovetzkii [22]. These authors obtain the same type of factorization theorem via unitary colligation methods as we obtain via operator algebra methods, but they assume somewhat different (and in many cases stronger) hypotheses.…”

Section: Introductionmentioning

confidence: 78%

See 1 more Smart Citation

Operator algebras of functions

Mittal

Paulsen

2010

Journal of Functional Analysis

View full text Add to dashboard Cite

We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite-dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.

show abstract

Section: Introductionmentioning

confidence: 78%

“…Applying our results, we obtain a factorization result for half planes. These algebras have been studied by D. KalyuzhnyiVerbovetzkii in [22]. where w(T ) denotes the numerical radius of T .…”

mentioning

confidence: 98%

Operator algebras of functions

Mittal

Paulsen

2010

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

“…If we allow QðzÞ ¼ bðzÞ aðzÞ ; this setting fits into our scheme. More generally, one can let so that D Q is a Cartesian product of half planes; this is the setting of recent work of Kalyuzhnyı˘-Verbovetzkiı˘ [32].…”

Section: Examples Of Domains D Qmentioning

confidence: 98%

Realization and interpolation for Schur–Agler-class functions on domains with matrix polynomial defining function in Cn

Ball

Bolotnikov

2004

Journal of Functional Analysis

113

View full text Add to dashboard Cite

We consider a bitangential interpolation problem for operator-valued functions defined on a general class of domains in C n (including as particular cases, Cartan domains of types I-III) which satisfy a type of von Neumann inequality associated with the domain. We show that any such function has a realization in terms of a unitary colligation and the defining polynomial for the domain. We show how the solution of various classes of bitangential interpolation problems for this class of functions corresponds to a unitary extension of a particular partially defined isometry uniquely specified by the interpolation data. Criteria for existence of solutions are given (1) in terms of positivity of a certain kernel completely determined by the data, or, more generally, (2) by the existence of a positive-kernel solution of a certain generalized Stein equation completely determined by the data. r 2004 Elsevier Inc. All rights reserved.

show abstract

“…For the Π d -Herglotz-Agler class, we are able to overcome the difficulty only in a special case (associated with the imposition of a growth condition at infinity), thereby recovering parallel results from [3]. For the most general Π d -Herglotz-Agler function f , we follow the LFT change-of-variable approach of [4] combined with the more general realization formalism (Schur complement of an operator pencil) suggested by the work of Bessmertnyȋ (see [18,19,20,21,28]) to arrive at a realization formula for the most general Π d -Herglotz-Agler function. We note that the original Bessmertnyȋ class involved additional symmetries leading to strong rigidity results.…”

Section: Introductionmentioning

confidence: 99%

“…We shall have occasion to need a Cayley transform (with both scalar and operator argument) acting between the right halfplane and the unit disk. Following the conventions in [28] and [11], we shall make use of the following version:…”

Section: Introductionmentioning

confidence: 99%

Schur–Agler and Herglotz–Agler classes of functions: Positive-kernel decompositions and transfer-function realizations

Ball

Kaliuzhnyi-Verbovetskyi

2015

Advances in Mathematics

View full text Add to dashboard Cite

Abstract. We discuss transfer-function realization for multivariable holomorphic functions mapping the unit polydisk or the right polyhalfplane into the operator analogue of either the unit disk or the right halfplane (Schur/Herglotz functions over either the unit polydisk or the right polyhalfplane) which satisfy the appropriate stronger contractive/positive real part condition for the values of these functions on commutative tuples of strict contractions/strictly accretive operators (Schur-Agler/Herglotz-Agler functions over either the unit polydisk or the right polyhalfplane). As originally shown by Agler, the first case (polydisk to disk) can be solved via unitary extensions of a partially defined isometry constructed in a canonical way from a kernel decomposition for the function (the lurking-isometry method). We show how a geometric reformulation of the lurking-isometry method (embedding of a given isotropic subspace of a Kreȋn space into a Lagrangian subspace-the lurking-isotropicsubspace method) can be used to handle the second two cases (polydisk to halfplane and polyhalfplane to disk), as well as the last case (polyhalfplane to halfplane) if an additional growth condition at ∞ is imposed. For the general fourth case, we show how a linear-fractional-transformation change of variable can be used to arrive at the appropriate symmetrized nonhomogeneous Bessmertnyȋ long-resolvent realization. We also indicate how this last result recovers the classical integral representation formula for scalar-valued holomorphic functions mapping the right halfplane into itself.

show abstract

On the Bessmertnyĭ Class of Homogeneous Positive Holomorphic Functions of Several Variables

Cited by 11 publications

References 16 publications

Operator algebras of functions

Operator algebras of functions

Realization and interpolation for Schur–Agler-class functions on domains with matrix polynomial defining function in Cn

Schur–Agler and Herglotz–Agler classes of functions: Positive-kernel decompositions and transfer-function realizations

Contact Info

Product

Resources

About