The Experts below are selected from a list of 306 Experts worldwide ranked by ideXlab platform
David P. Kimsey - One of the best experts on this subject based on the ideXlab platform.
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Spectral Theorem for Unitary Operators
Spectral Theory on the S-Spectrum for Quaternionic Operators, 2020Co-Authors: Fabrizio Colombo, Jonathan Gantner, David P. KimseyAbstract:The Spectral Theorem for unitary operators is a particular case of the Spectral Theorem for bounded normal operators proved in Chapter 11. However, as in the complex case, the Spectral Theorem for unitary operators can be deduced from the quaternionic version of Herglotz’s Theorem proved in [16]. The Spectral Theorem for unitary operators based on Herglotz’s Theorem was proved in [14].
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The Spectral Theorem for bounded normal operators
Spectral Theory on the S-Spectrum for Quaternionic Operators, 2020Co-Authors: Fabrizio Colombo, Jonathan Gantner, David P. KimseyAbstract:In this chapter we prove the Spectral Theorem for bounded normal operators T in \(\mathcal{B}(\mathcal{H})\).
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the Spectral Theorem for unitary operators based on the s spectrum
Milan Journal of Mathematics, 2016Co-Authors: Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene SabadiniAbstract:The quaternionic Spectral Theorem has already been considered in the literature, see e.g. [22], [32], [33], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the Theorem.
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the Spectral Theorem for quaternionic unbounded normal operators based on the s spectrum
Journal of Mathematical Physics, 2016Co-Authors: Daniel Alpay, Fabrizio Colombo, David P. KimseyAbstract:In this paper we prove the Spectral Theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a Spectral Theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of Spectral analysis of quaternionic operators. The S-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the Spectral Theorem for quaternionic normal operators. The lack of a suitable notion of spectrum was a major obstruction to fully understand the Spectral Theorem for quaternionic normal operators. A prime motivation for studying the Spectral Theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.
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the Spectral Theorem for quaternionic unbounded normal operators based on the s spectrum
arXiv: Spectral Theory, 2014Co-Authors: Daniel Alpay, Fabrizio Colombo, David P. KimseyAbstract:In this paper we prove the Spectral Theorem for quaternionic unbounded normal operators using the notion of $S$-spectrum. The proof technique consists of first establishing a Spectral Theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of Spectral analysis of quaternionic operators. The $S$-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the Spectral Theorem for quaternionic normal operators. The fact that the correct notion of spectrum for quaternionic operators was not previously known has been one of the main obstructions to fully understanding the Spectral Theorem in this setting. A prime motivation for studying the Spectral Theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.
Fabrizio Colombo - One of the best experts on this subject based on the ideXlab platform.
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Spectral Theorem for Unitary Operators
Spectral Theory on the S-Spectrum for Quaternionic Operators, 2020Co-Authors: Fabrizio Colombo, Jonathan Gantner, David P. KimseyAbstract:The Spectral Theorem for unitary operators is a particular case of the Spectral Theorem for bounded normal operators proved in Chapter 11. However, as in the complex case, the Spectral Theorem for unitary operators can be deduced from the quaternionic version of Herglotz’s Theorem proved in [16]. The Spectral Theorem for unitary operators based on Herglotz’s Theorem was proved in [14].
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The Spectral Theorem for bounded normal operators
Spectral Theory on the S-Spectrum for Quaternionic Operators, 2020Co-Authors: Fabrizio Colombo, Jonathan Gantner, David P. KimseyAbstract:In this chapter we prove the Spectral Theorem for bounded normal operators T in \(\mathcal{B}(\mathcal{H})\).
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the Spectral Theorem for unitary operators based on the s spectrum
Milan Journal of Mathematics, 2016Co-Authors: Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene SabadiniAbstract:The quaternionic Spectral Theorem has already been considered in the literature, see e.g. [22], [32], [33], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the Theorem.
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the Spectral Theorem for quaternionic unbounded normal operators based on the s spectrum
Journal of Mathematical Physics, 2016Co-Authors: Daniel Alpay, Fabrizio Colombo, David P. KimseyAbstract:In this paper we prove the Spectral Theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a Spectral Theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of Spectral analysis of quaternionic operators. The S-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the Spectral Theorem for quaternionic normal operators. The lack of a suitable notion of spectrum was a major obstruction to fully understand the Spectral Theorem for quaternionic normal operators. A prime motivation for studying the Spectral Theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.
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the Spectral Theorem for quaternionic unbounded normal operators based on the s spectrum
arXiv: Spectral Theory, 2014Co-Authors: Daniel Alpay, Fabrizio Colombo, David P. KimseyAbstract:In this paper we prove the Spectral Theorem for quaternionic unbounded normal operators using the notion of $S$-spectrum. The proof technique consists of first establishing a Spectral Theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of Spectral analysis of quaternionic operators. The $S$-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the Spectral Theorem for quaternionic normal operators. The fact that the correct notion of spectrum for quaternionic operators was not previously known has been one of the main obstructions to fully understanding the Spectral Theorem in this setting. A prime motivation for studying the Spectral Theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.
Daniel Alpay - One of the best experts on this subject based on the ideXlab platform.
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the Spectral Theorem for unitary operators based on the s spectrum
Milan Journal of Mathematics, 2016Co-Authors: Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene SabadiniAbstract:The quaternionic Spectral Theorem has already been considered in the literature, see e.g. [22], [32], [33], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the Theorem.
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the Spectral Theorem for quaternionic unbounded normal operators based on the s spectrum
Journal of Mathematical Physics, 2016Co-Authors: Daniel Alpay, Fabrizio Colombo, David P. KimseyAbstract:In this paper we prove the Spectral Theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a Spectral Theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of Spectral analysis of quaternionic operators. The S-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the Spectral Theorem for quaternionic normal operators. The lack of a suitable notion of spectrum was a major obstruction to fully understand the Spectral Theorem for quaternionic normal operators. A prime motivation for studying the Spectral Theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.
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the Spectral Theorem for quaternionic unbounded normal operators based on the s spectrum
arXiv: Spectral Theory, 2014Co-Authors: Daniel Alpay, Fabrizio Colombo, David P. KimseyAbstract:In this paper we prove the Spectral Theorem for quaternionic unbounded normal operators using the notion of $S$-spectrum. The proof technique consists of first establishing a Spectral Theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of Spectral analysis of quaternionic operators. The $S$-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the Spectral Theorem for quaternionic normal operators. The fact that the correct notion of spectrum for quaternionic operators was not previously known has been one of the main obstructions to fully understanding the Spectral Theorem in this setting. A prime motivation for studying the Spectral Theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.
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the Spectral Theorem for unitary operators based on the s spectrum
arXiv: Spectral Theory, 2014Co-Authors: Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene SabadiniAbstract:The quaternionic Spectral Theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the Theorem. In this paper we prove the quaternionic Spectral Theorem for unitary operators using the $S$-spectrum. In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known Theorem for matrices. The notion of $S$-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for $n$-tuples of not necessarily commuting operators, to define and study a noncommutative versions of the Riesz-Dunford functional calculus. The main tools to prove the Spectral Theorem for unitary operators are the quaternionic version of Herglotz's Theorem, which relies on the new notion of $q$-positive measure, and quaternionic Spectral measures, which are related to the quaternionic Riesz projectors defined by means of the $S$-resolvent operator and the $S$-spectrum. The results in this paper restore the analogy with the complex case in which the classical notion of spectrum appears in the Riesz-Dunford functional calculus as well as in the Spectral Theorem.
David Krejciřik - One of the best experts on this subject based on the ideXlab platform.
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calculation of the metric in the hilbert space of a cal p cal t symmetric model via the Spectral Theorem
Journal of Physics A, 2008Co-Authors: David KrejciřikAbstract:In a previous paper [1] we introduced a very simple -symmetric non-Hermitian Hamiltonian with a real spectrum and derived a closed formula for the metric operator relating the problem to a Hermitian one. In this paper we propose an alternative formula for the metric operator, which we believe is more elegant and whose construction—based on a backward use of the Spectral Theorem for self-adjoint operators—provides new insights into the nature of the model.
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Calculation of the metric in the Hilbert space of a {\cal P}{\cal T} -symmetric model via the Spectral Theorem
Journal of Physics A, 2008Co-Authors: David KrejciřikAbstract:In a previous paper [1] we introduced a very simple -symmetric non-Hermitian Hamiltonian with a real spectrum and derived a closed formula for the metric operator relating the problem to a Hermitian one. In this paper we propose an alternative formula for the metric operator, which we believe is more elegant and whose construction—based on a backward use of the Spectral Theorem for self-adjoint operators—provides new insights into the nature of the model.
Tomaž Prosen - One of the best experts on this subject based on the ideXlab platform.
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Spectral Theorem for the lindblad equation for quadratic open fermionic systems
Journal of Statistical Mechanics: Theory and Experiment, 2010Co-Authors: Tomaž ProsenAbstract:The Spectral Theorem is proven for the quantum dynamics of quadratic open systems of n fermions described by the Lindblad equation. Invariant eigenspaces of the many-body Liouvillian dynamics and their largest Jordan blocks are explicitly constructed for all eigenvalues. For eigenvalue zero we describe an algebraic procedure for constructing (possibly higher dimensional) spaces of (degenerate) non-equilibrium steady states.
