The Experts below are selected from a list of 19686 Experts worldwide ranked by ideXlab platform
Irene Sabadini - One of the best experts on this subject based on the ideXlab platform.
-
the h Functional Calculus based on the s spectrum for quaternionic operators and for n tuples of noncommuting operators
Journal of Functional Analysis, 2016Co-Authors: Daniel Alpay, F Colombo, Tao Qian, Irene SabadiniAbstract:Abstract In this paper we extend the H ∞ Functional Calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated Functional Calculus, called S-Functional Calculus. The S-Functional Calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford Functional Calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S-Functional Calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H ∞ Functional Calculus based on the notion of S-spectrum for both quaternionic operators and for n-tuples of noncommuting operators. We remark that the H ∞ Functional Calculus for ( n + 1 ) -tuples of operators applies, in particular, to the Dirac operator.
-
The H∞ Functional Calculus Based on the S-Spectrum for Quaternionic Operators and for N-Tuples of Noncommuting Operators
Journal of Functional Analysis, 2016Co-Authors: Daniel Alpay, Fabrizio Colombo, Tao Qian, Irene SabadiniAbstract:Abstract In this paper we extend the H ∞ Functional Calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated Functional Calculus, called S-Functional Calculus. The S-Functional Calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford Functional Calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S-Functional Calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H ∞ Functional Calculus based on the notion of S-spectrum for both quaternionic operators and for n-tuples of noncommuting operators. We remark that the H ∞ Functional Calculus for ( n + 1 ) -tuples of operators applies, in particular, to the Dirac operator.
-
The $H^\infty$ Functional Calculus based on the $S$-spectrum for quaternionic operators and for $n$-tuples of noncommuting operators
arXiv: Functional Analysis, 2015Co-Authors: Daniel Alpay, Fabrizio Colombo, Tao Qian, Irene SabadiniAbstract:In this paper we extend the $H^\infty$ Functional Calculus to quaternionic operators and to $n$-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated Functional Calculus, called $S$-Functional Calculus. The $S$-Functional Calculus has two versions one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz-Dunford Functional Calculus based on slice hyperholomorphicity because it shares with it the most important properties. The $S$-Functional Calculus is based on the notion of $S$-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the $H^\infty$ Functional Calculus based on the notion of $S$-spectrum for both quaternionic operators and for $n$-tuples of noncommuting operators. We remark that the $H^\infty$ Functional Calculus for $(n+1)$-tuples of operators applies, in particular, to the Dirac operator.
-
A New Resolvent Equation for the S-Functional Calculus
The Journal of Geometric Analysis, 2014Co-Authors: Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, Irene SabadiniAbstract:The \(S\)-Functional Calculus is a Functional Calculus for \((n+1)\)-tuples of not necessarily commuting operators that can be considered a higher-dimensional version of the classical Riesz–Dunford Functional Calculus for a single operator. In this last Calculus, the resolvent equation plays an important role in the proof of several results. Associated with the \(S\)-Functional Calculus there are two resolvent operators: the left \(S_L^{-1}(s,T)\) and the right one \(S_R^{-1}(s,T)\), where \(s=(s_0,s_1,\ldots ,s_n)\in \mathbb {R}^{n+1}\) and \(T=(T_0,T_1,\ldots ,T_n)\) is an \((n+1)\)-tuple of noncommuting operators. The two \(S\)-resolvent operators satisfy the \(S\)-resolvent equations \(S_L^{-1}(s,T)s-TS_L^{-1}(s,T)=\mathcal {I}\), and \(sS_R^{-1}(s,T)-S_R^{-1}(s,T)T=\mathcal {I}\), respectively, where \(\mathcal {I}\) denotes the identity operator. These equations allow us to prove some properties of the \(S\)-Functional Calculus. In this paper we prove a new resolvent equation which is the analog of the classical resolvent equation. It is interesting to note that the equation involves both the left and the right \(S\)-resolvent operators simultaneously.
-
Bicomplex Holomorphic Functional Calculus
Mathematische Nachrichten, 2013Co-Authors: Fabrizio Colombo, Irene Sabadini, Daniele C. StruppaAbstract:In this paper we introduce and study a Functional Calculus for bicomplex linear bounded operators. The study is based on the decomposition of bicomplex numbers and of linear operators using the two nonreal idempotents. We show that, due to the presence of zero divisors in the bicomplex numbers, the spectrum of a bounded operator is unbounded. We therefore introduce a different spectrum (called reduced spectrum) which is bounded and turns out to be the right tool to construct the bicomplex holomorphic Functional Calculus. Finally we provide some properties of the Calculus.
Fabrizio Colombo - One of the best experts on this subject based on the ideXlab platform.
-
The Phillips Functional Calculus
Quaternionic Closed Operators Fractional Powers and Fractional Diffusion Processes, 2019Co-Authors: Fabrizio Colombo, Jonathan GantnerAbstract:In Chapter 3, we have introduced the direct approach to the S-Functional Calculus for unbounded operators, which only requires the operator T to be closed and have a nonempty S-resolvent set.
-
The F-Functional Calculus for Bounded Operators
Spectral Theory on the S-Spectrum for Quaternionic Operators, 2018Co-Authors: Fabrizio Colombo, Jonathan Gantner, David P. KimseyAbstract:The Fueter mapping theorem in integral form introduced in [86], see Chapter 2.2, provides an integral transform that turns slice hyperholomorphic functions into Fueter regular ones. By formally replacing the scalar variable in this integral transform by an operator T, we obtain a Functional Calculus for Fueter regular functions that is based on the theory of slice hyperholomorphic functions. The F-Functional Calculus was introduced and studied in the following papers [54, 78, 81, 86].
-
The H∞-Functional Calculus
Spectral Theory on the S-Spectrum for Quaternionic Operators, 2018Co-Authors: Fabrizio Colombo, Jonathan Gantner, David P. KimseyAbstract:The H∞-Functional Calculus is an extension of the Riesz-Dunford Functional Calculus for bounded operators to unbounded sectorial operators, and it was introduced by A. McIntosh in [165]; see also [5]. This Calculus is connected with pseudodifferential operators, with Kato’s square root problem, and with the study of evolution equations and, in particular, the characterization of maximal regularity and with the fractional powers of differential operators. For an overview and more problems associated with this Functional Calculus for the classical case, see the book [156] and the references therein.
-
Properties of the S-Functional Calculus for bounded operators
Spectral Theory on the S-Spectrum for Quaternionic Operators, 2018Co-Authors: Fabrizio Colombo, Jonathan Gantner, David P. KimseyAbstract:In this chapter we will show that most of the properties that hold for the Riesz-Dunford Functional Calculus can be extended to the S-Functional Calculus. The proofs of the quaternionic results require several additional efforts with respect to the classical case.
-
The H∞ Functional Calculus Based on the S-Spectrum for Quaternionic Operators and for N-Tuples of Noncommuting Operators
Journal of Functional Analysis, 2016Co-Authors: Daniel Alpay, Fabrizio Colombo, Tao Qian, Irene SabadiniAbstract:Abstract In this paper we extend the H ∞ Functional Calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated Functional Calculus, called S-Functional Calculus. The S-Functional Calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford Functional Calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S-Functional Calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H ∞ Functional Calculus based on the notion of S-spectrum for both quaternionic operators and for n-tuples of noncommuting operators. We remark that the H ∞ Functional Calculus for ( n + 1 ) -tuples of operators applies, in particular, to the Dirac operator.
Daniele C. Struppa - One of the best experts on this subject based on the ideXlab platform.
-
Bicomplex Holomorphic Functional Calculus
Mathematische Nachrichten, 2013Co-Authors: Fabrizio Colombo, Irene Sabadini, Daniele C. StruppaAbstract:In this paper we introduce and study a Functional Calculus for bicomplex linear bounded operators. The study is based on the decomposition of bicomplex numbers and of linear operators using the two nonreal idempotents. We show that, due to the presence of zero divisors in the bicomplex numbers, the spectrum of a bounded operator is unbounded. We therefore introduce a different spectrum (called reduced spectrum) which is bounded and turns out to be the right tool to construct the bicomplex holomorphic Functional Calculus. Finally we provide some properties of the Calculus.
-
noncommutative Functional Calculus theory and applications of slice hyperholomorphic functions
2011Co-Authors: F Colombo, Irene Sabadini, Daniele C. StruppaAbstract:1 Introduction.- 2 Slice monogenic functions.- 3 Functional Calculus for n-tuples of operators.- 4 Quaternionic Functional Calculus.- 5 Appendix: The Riesz-Dunford Functional Calculus.- Bibliography.- Index.
-
Appendix: The Riesz–Dunford Functional Calculus
Noncommutative Functional Calculus, 2011Co-Authors: Fabrizio Colombo, Irene Sabadini, Daniele C. StruppaAbstract:In this Appendix we collect some basic material on the Riesz–Dunford Functional Calculus useful for the readers who are not familiar with this subject. This background, with all the details, can be found in [35] and [91].
-
non commutative Functional Calculus bounded operators
Complex Analysis and Operator Theory, 2010Co-Authors: F Colombo, Irene Sabadini, Graziano Gentili, Daniele C. StruppaAbstract:In this paper we develop a Functional Calculus for bounded operators defined on quaternionic Banach spaces. This Calculus is based on the new notion of slice-regularity, see Gentili and Struppa (Acad Sci Paris 342:741–744, 2006) and the key tools are a new resolvent operator and a new eigenvalue problem.
-
Non-commutative Functional Calculus: Unbounded operators
Journal of Geometry and Physics, 2010Co-Authors: Fabrizio Colombo, Irene Sabadini, Graziano Gentili, Daniele C. StruppaAbstract:In a recent work, \cite{cgss}, we developed a Functional Calculus for bounded operators defined on quaternionic Banach spaces. In this paper we show how the results from \cite{cgss} can be extended to the unbounded case, and we highlight the crucial differences between the two cases. In particular, we deduce a new eigenvalue equation, suitable for the construction of a Functional Calculus for operators whose spectrum is not necessarily real
Daniel Alpay - One of the best experts on this subject based on the ideXlab platform.
-
the h Functional Calculus based on the s spectrum for quaternionic operators and for n tuples of noncommuting operators
Journal of Functional Analysis, 2016Co-Authors: Daniel Alpay, F Colombo, Tao Qian, Irene SabadiniAbstract:Abstract In this paper we extend the H ∞ Functional Calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated Functional Calculus, called S-Functional Calculus. The S-Functional Calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford Functional Calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S-Functional Calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H ∞ Functional Calculus based on the notion of S-spectrum for both quaternionic operators and for n-tuples of noncommuting operators. We remark that the H ∞ Functional Calculus for ( n + 1 ) -tuples of operators applies, in particular, to the Dirac operator.
-
The H∞ Functional Calculus Based on the S-Spectrum for Quaternionic Operators and for N-Tuples of Noncommuting Operators
Journal of Functional Analysis, 2016Co-Authors: Daniel Alpay, Fabrizio Colombo, Tao Qian, Irene SabadiniAbstract:Abstract In this paper we extend the H ∞ Functional Calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated Functional Calculus, called S-Functional Calculus. The S-Functional Calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford Functional Calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S-Functional Calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H ∞ Functional Calculus based on the notion of S-spectrum for both quaternionic operators and for n-tuples of noncommuting operators. We remark that the H ∞ Functional Calculus for ( n + 1 ) -tuples of operators applies, in particular, to the Dirac operator.
-
The $H^\infty$ Functional Calculus based on the $S$-spectrum for quaternionic operators and for $n$-tuples of noncommuting operators
arXiv: Functional Analysis, 2015Co-Authors: Daniel Alpay, Fabrizio Colombo, Tao Qian, Irene SabadiniAbstract:In this paper we extend the $H^\infty$ Functional Calculus to quaternionic operators and to $n$-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated Functional Calculus, called $S$-Functional Calculus. The $S$-Functional Calculus has two versions one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz-Dunford Functional Calculus based on slice hyperholomorphicity because it shares with it the most important properties. The $S$-Functional Calculus is based on the notion of $S$-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the $H^\infty$ Functional Calculus based on the notion of $S$-spectrum for both quaternionic operators and for $n$-tuples of noncommuting operators. We remark that the $H^\infty$ Functional Calculus for $(n+1)$-tuples of operators applies, in particular, to the Dirac operator.
-
A New Resolvent Equation for the S-Functional Calculus
The Journal of Geometric Analysis, 2014Co-Authors: Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, Irene SabadiniAbstract:The \(S\)-Functional Calculus is a Functional Calculus for \((n+1)\)-tuples of not necessarily commuting operators that can be considered a higher-dimensional version of the classical Riesz–Dunford Functional Calculus for a single operator. In this last Calculus, the resolvent equation plays an important role in the proof of several results. Associated with the \(S\)-Functional Calculus there are two resolvent operators: the left \(S_L^{-1}(s,T)\) and the right one \(S_R^{-1}(s,T)\), where \(s=(s_0,s_1,\ldots ,s_n)\in \mathbb {R}^{n+1}\) and \(T=(T_0,T_1,\ldots ,T_n)\) is an \((n+1)\)-tuple of noncommuting operators. The two \(S\)-resolvent operators satisfy the \(S\)-resolvent equations \(S_L^{-1}(s,T)s-TS_L^{-1}(s,T)=\mathcal {I}\), and \(sS_R^{-1}(s,T)-S_R^{-1}(s,T)T=\mathcal {I}\), respectively, where \(\mathcal {I}\) denotes the identity operator. These equations allow us to prove some properties of the \(S\)-Functional Calculus. In this paper we prove a new resolvent equation which is the analog of the classical resolvent equation. It is interesting to note that the equation involves both the left and the right \(S\)-resolvent operators simultaneously.
-
A new resolvent equation for the S-Functional Calculus
arXiv: Functional Analysis, 2013Co-Authors: Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, Irene SabadiniAbstract:The S-Functional Calculus is a Functional Calculus for $(n+1)$-tuples of non necessarily commuting operators that can be considered a higher dimensional version of the classical Riesz-Dunford Functional Calculus for a single operator. In this last Calculus, the resolvent equation plays an important role in the proof of several results. Associated with the S-Functional Calculus there are two resolvent operators: the left $S_L^{-1}(s,T)$ and the right one $S_R^{-1}(s,T)$, where $s=(s_0,s_1,\ldots,s_n)\in \mathbb{R}^{n+1}$ and $T=(T_0,T_1,\ldots,T_n)$ is an $(n+1)$-tuple of non commuting operators. These two S-resolvent operators satisfy the S-resolvent equations $S_L^{-1}(s,T)s-TS_L^{-1}(s,T)=\mathcal{I}$, and $sS_R^{-1}(s,T)-S_R^{-1}(s,T)T=\mathcal{I}$, respectively, where $\mathcal{I}$ denotes the identity operator. These equations allows to prove some properties of the S-Functional Calculus. In this paper we prove a new resolvent equation for the S-Functional Calculus which is the analogue of the classical resolvent equation. It is interesting to note that the equation involves both the left and the right S-resolvent operators simultaneously.
A. R. Mirotin - One of the best experts on this subject based on the ideXlab platform.
-
On the connections of Hille-Phillips Functional Calculus with Bochner-Phillips Functional Calculus
arXiv: Functional Analysis, 2019Co-Authors: A. R. MirotinAbstract:The extension of Hille-Phillips Functional Calculus of semigroup generators which leads to unbounded operators is considered. Connections of this Calculus to Bochner-Phillips Functional Calculus are indicated. In particular, the multiplication rule and the composition rule are proved. Several examples are given.
-
The unbounded extension of Hille-Phillips Functional Calculus
arXiv: Functional Analysis, 2019Co-Authors: A. R. MirotinAbstract:The extension of Hille-Phillips Functional Calculus of semigroup generators which leads to unbounded operators is given. Connections of this Calculus to Bochner-Phillips Functional Calculus are indicated, and several examples are considered.
-
On multidimensional Bochner-Phillips Functional Calculus
arXiv: Functional Analysis, 2019Co-Authors: A. R. MirotinAbstract:The Functional Calculus of semigroup generators, based on the class of Bernstein functions in several variables is developed, the condition for holomorphy of semigroups, generated by operators which arisen in the Calculus is given, and in the one-dimensional case the moment inequality for such operators is proved.
-
On some Functional Calculus of closed operators in a Banach space. II
Russian Mathematics, 2015Co-Authors: A. A. Atvinovskii, A. R. MirotinAbstract:We develop a Functional Calculus for a class of closed operators in a Banach space. We establish several properties of the Calculus under consideration such as continuity, stability, and uniqueness theorems, the spectral mapping theorem, the inversion theorem, and some others. In our preceding work we developed an analogous Functional Calculus for a narrower class of operators.
