Generalized Sunder inequality

Sangeeta Jhanjee; Alan James Pryde

doi:10.1007/978-3-319-18182-0_5

Generalized Sunder inequality

Sangeeta Jhanjee, Alan James Pryde

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference Paper › Research › peer-review

Abstract

V. Sunder proved that for n×n complex matrices A and B, with A being Hermitian and B being skew Hermitian with eigenvalues {αi}ni=1{αi}i=1n and {βi}ni=1{βi}i=1n respectively (counting multiplicity) such that
|α1|≥⋯≥|αn|,|β1|≤⋯≤|βn|
|α1|≥⋯≥|αn|,|β1|≤⋯≤|βn|
then
|αi−βi|≤∥A−B∥
|αi−βi|≤∥A−B∥
where ∥⋅∥∥⋅∥ is the operator bound norm. We generalize Sunder’s result to the case of an m-tuple of n × n complex matrices, using the Clifford operator.

Original language	English
Title of host publication	Operator Algebras and Mathematical Physics: 24th International Workshop in Operator Theory and its Applications, Bangalore, December 2013 (IWOTA 2013)
Editors	Tirthankar Bhattacharyya, Michael A Dritschel
Place of Publication	Cham Switzerland
Publisher	Springer
Pages	83-86
Number of pages	4
Volume	247
ISBN (Electronic)	9783319181820
ISBN (Print)	9783319181813
DOIs	https://doi.org/10.1007/978-3-319-18182-0_5
Publication status	Published - 2015
Event	International Workshop on Operator Theory and its Applications (IWOTA) 2013 - Indian Institute of Science, Bangalore, India Duration: 16 Dec 2013 → 20 Dec 2013 Conference number: 24th http://math.iisc.ernet.in/~iwota2013/

Workshop

Workshop	International Workshop on Operator Theory and its Applications (IWOTA) 2013
Abbreviated title	IWOTA 2013
Country	India
City	Bangalore
Period	16/12/13 → 20/12/13
Internet address	http://math.iisc.ernet.in/~iwota2013/

Keywords

Joint eigenvalues
joint spectra
Clifford algebra
commuting tuple of matrices
Sunder’s inequality

Access to Document

10.1007/978-3-319-18182-0_5

Cite this

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title = "Generalized Sunder inequality",

abstract = "V. Sunder proved that for n×n complex matrices A and B, with A being Hermitian and B being skew Hermitian with eigenvalues {αi}ni=1{αi}i=1n and {βi}ni=1{βi}i=1n respectively (counting multiplicity) such that|α1|≥⋯≥|αn|,|β1|≤⋯≤|βn||α1|≥⋯≥|αn|,|β1|≤⋯≤|βn|then|αi−βi|≤∥A−B∥|αi−βi|≤∥A−B∥where ∥⋅∥∥⋅∥ is the operator bound norm. We generalize Sunder{\textquoteright}s result to the case of an m-tuple of n × n complex matrices, using the Clifford operator.",

keywords = "Joint eigenvalues, joint spectra, Clifford algebra, commuting tuple of matrices, Sunder{\textquoteright}s inequality",

author = "Sangeeta Jhanjee and Pryde, {Alan James}",

year = "2015",

doi = "10.1007/978-3-319-18182-0_5",

language = "English",

isbn = "9783319181813",

volume = "247",

pages = "83--86",

editor = "Tirthankar Bhattacharyya and Dritschel, {Michael A}",

booktitle = "Operator Algebras and Mathematical Physics: 24th International Workshop in Operator Theory and its Applications, Bangalore, December 2013 (IWOTA 2013)",

publisher = "Springer",

note = "International Workshop on Operator Theory and its Applications (IWOTA) 2013, IWOTA 2013 ; Conference date: 16-12-2013 Through 20-12-2013",

url = "http://math.iisc.ernet.in/~iwota2013/",

}

Jhanjee, S & Pryde, AJ 2015, Generalized Sunder inequality. in T Bhattacharyya & MA Dritschel (eds), Operator Algebras and Mathematical Physics: 24th International Workshop in Operator Theory and its Applications, Bangalore, December 2013 (IWOTA 2013). vol. 247, Springer, Cham Switzerland, pp. 83-86, International Workshop on Operator Theory and its Applications (IWOTA) 2013, Bangalore, India, 16/12/13. https://doi.org/10.1007/978-3-319-18182-0_5

Generalized Sunder inequality. / Jhanjee, Sangeeta; Pryde, Alan James.

Operator Algebras and Mathematical Physics: 24th International Workshop in Operator Theory and its Applications, Bangalore, December 2013 (IWOTA 2013). ed. / Tirthankar Bhattacharyya; Michael A Dritschel. Vol. 247 Cham Switzerland : Springer, 2015. p. 83-86.

Research output: Chapter in Book/Report/Conference proceeding › Conference Paper › Research › peer-review

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T1 - Generalized Sunder inequality

AU - Jhanjee, Sangeeta

AU - Pryde, Alan James

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PY - 2015

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N2 - V. Sunder proved that for n×n complex matrices A and B, with A being Hermitian and B being skew Hermitian with eigenvalues {αi}ni=1{αi}i=1n and {βi}ni=1{βi}i=1n respectively (counting multiplicity) such that|α1|≥⋯≥|αn|,|β1|≤⋯≤|βn||α1|≥⋯≥|αn|,|β1|≤⋯≤|βn|then|αi−βi|≤∥A−B∥|αi−βi|≤∥A−B∥where ∥⋅∥∥⋅∥ is the operator bound norm. We generalize Sunder’s result to the case of an m-tuple of n × n complex matrices, using the Clifford operator.

AB - V. Sunder proved that for n×n complex matrices A and B, with A being Hermitian and B being skew Hermitian with eigenvalues {αi}ni=1{αi}i=1n and {βi}ni=1{βi}i=1n respectively (counting multiplicity) such that|α1|≥⋯≥|αn|,|β1|≤⋯≤|βn||α1|≥⋯≥|αn|,|β1|≤⋯≤|βn|then|αi−βi|≤∥A−B∥|αi−βi|≤∥A−B∥where ∥⋅∥∥⋅∥ is the operator bound norm. We generalize Sunder’s result to the case of an m-tuple of n × n complex matrices, using the Clifford operator.

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KW - joint spectra

KW - Clifford algebra

KW - commuting tuple of matrices

KW - Sunder’s inequality

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DO - 10.1007/978-3-319-18182-0_5

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BT - Operator Algebras and Mathematical Physics: 24th International Workshop in Operator Theory and its Applications, Bangalore, December 2013 (IWOTA 2013)

A2 - Bhattacharyya, Tirthankar

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PB - Springer

CY - Cham Switzerland

T2 - International Workshop on Operator Theory and its Applications (IWOTA) 2013

Y2 - 16 December 2013 through 20 December 2013

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