Numerical Shadow

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Numerical range

General

1. [1] 2. [2] 3. [3] 4. [4] 5. [5] 6. [6] 7. [7] 8. [8] 9. [9] 10. [10] 11. [11] 12. [12] 13. [13] 14. [14] 15. [15] 16. [16] 17. [17] 18. [14] 19. [18] 20. [19] 21. [20] 22. [21] 23. [22] 24. [23] 25. [24] 26. [25] 27. [26] 28. [27] 29. [28] 30. [29] 31. [30] 32. [31] 33. [32] 34. [33]

1. O. Toeplitz, 1918. Das algebraische Analogon zu einem Satze von Fejer. Mathematische Zeitschrift, 2, Springer, pp.187–197.
2. F. Hausdorff, 1919. Der Wertevorrat einer Bilinearform. Mathematische Zeitschrift, 3, Springer, pp.314–316.
3. F. D Murnaghan, 1932. On the field of values of a square matrix. Proceedings of the National Academy of Sciences of the United States of America, 18, National Academy of Sciences, pp.246.
4. R. Kippenhahn, 1951. Uber den Wertevorrat einer matrix. Mathematische Nachrichten, 6, Wiley Online Library, pp.193–228.
5. R. Horn, C. Johnson, 1994. Topics in matrix analysis. Cambridge university press.
6. K. E. Gustafson, D. K. M. Rao, 1997. Numerical range: The Field of Values of Linear Operators and Matrices. Springer.
7. E. Gutkin, 2004. The Toeplitz-Hausdorff theorem revisited: relating linear algebra and geometry. The Mathematical Intelligencer, 26, Springer, pp.8–14.
8. C. K. Li, 1996. A simple proof of the elliptical range theorem. Proceedings of the American Mathematical Society, 124, pp.1985–1986.
9. D. S. Keeler, L. Rodman, I. M. Spitkovsky, 1997. The numerical range of 3x3 matrices. Linear Algebra and its Applications, 252, pp.115 - 139.
10. M. Crouzeix, 2006. Open problems on Numerical range and functional calculus.
11. C. R. Johnson, 1981. Numerical ranges of principal submatrices. Linear Algebra and its Applications, 1, pp.23-34.
12. C. R. Johnson, 1976. Normality and the numerical range. Linear Algebra and its Applications, 37, Elsevier, pp.89–94.
13. P. Nylen, T. Y. Tam, 1991. Numerical range of a doubly stochastic matrix. Linear Algebra and Its Applications, 153, Elsevier, pp.161–176.
14. Panayiotis J Psarrakos, Michael J Tsatsomeros, 2002. Numerical ​range:(in) a matrix nutshell. Department of Mathematics,​ Washington State University.
15. Ch. Chorianopoulos, S. Karanasios, P. Psarrakos, 2009. A definition of numerical range of rectangular matrices. Linear and Multilinear Algebra, 57, Taylor & Francis, pp.459–475.
16. J. H. Shapiro, 2004. Notes on the numerical range. Lecture Notes, Michigan State University.
17. P. Skoufranis, 2012. Numerical Ranges of Operators.
18. Russell Carden, 2009. A simple algorithm for the inverse field of values problem. Inverse Problems, 25, IOP Publishing, pp.115019.
19. Moshe Goldberg, Ernst Straus, 1977. On characterizations and integrals of generalized numerical ranges. Pacific Journal of Mathematics, 69, pp.45–54.
20. Moshe Goldberg, 1979. On certain finite dimensional numerical ranges and numerical radii. Linear and Multilinear Algebra, 7, Taylor & Francis, pp.329–342.
21. Joseph Stampfli, 1970. The norm of a derivation. Pacific journal of mathematics, 33, Mathematical Sciences Publishers, pp.737–747.
22. Hwa-Long Gau, Pei Yuan Wu, 2007. Numerical ranges of companion matrices. Linear algebra and its applications, 421, Elsevier, pp.202–218.
23. Brian Lins, 2020. Numerical ranges encircled by analytic curves. arXiv preprint arXiv:2003.05347.
24. Kelly Bickel, Pamela Gorkin, 2018. Numerical Range and Compressions of the Shift. arXiv preprint arXiv:1810.11680.
25. Jaedeok Kim, Youngmi Kim, 2018. Jordan Plane and Numerical Range of Operators Involving Two Projections. arXiv preprint arXiv:1811.10518.
26. Mahsa Fatehi, Asma Negahdari, 2019. Numerical range of weighted composition operators which contain zero. arXiv preprint arXiv:1901.07736.
27. Chi-Kwong Li, Yiu-Tung Poon, 2019. Numerical Range Inclusion, Dilation, and Operator Systems. arXiv preprint arXiv:1911.01221.
28. Titas Geryba, Ilya M Spitkovsky, 2020. On the numerical range of some block matrices with scalar diagonal blocks. Linear and Multilinear Algebra, Taylor and Francis, pp.1–14.
29. Nam-Kiu Tsin, 1983. Diameter and minimal width of the numerical range. Linear and multilinear algebra, 14, Taylor and Francis, pp.179–185.
30. Jean-Christophe Bourin, Antoine Mhanna, 2017. Positive block matrices and numerical ranges. Comptes Rendus Mathematique, 355, Elsevier, pp.1077–1081.
31. Mao-Ting Chien, Hiroshi Nakazato, Jie Meng, 2019. The diameter and width of numerical ranges. Linear Algebra and its Applications, 582, Elsevier, pp.76–98.
32. Kukulski Ryszard, Lewandowska Paulina, Pawela Łukasz, 2020. Perturbation of the numerical range of unitary matrices. arXiv preprint arXiv: 2002.05553v1.
33. Ilya M Spitkovsky, Stephan Weis, 2018. Signatures of quantum phase transitions from the boundary of the numerical range. Journal of mathematical physics, 59, AIP Publishing, pp.121901.

Geometry of numerical range

1. [34] 2. [35] 3. [36] 4. [37] 5. [38] 6. [39] 7. [40] 8. [41] 9. [42] 10. [43]

34. M. Fiedler, 1981. Geometry of the numerical range of matrices. Linear Algebra and its Applications, 37, Elsevier, pp.81–96.
35. E. A. Jonckheere, F. Ahmad, E. Gutkin, 1998. Differential topology of numerical range. Linear algebra and its applications, 279, Elsevier, pp.227–254.
36. D. Henrion, 2010. Semidefinite geometry of the numerical range. Electronic Journal of Linear Algebra, 20, pp.322-332.
37. J. W. Helton, I. M. Spitkovsky, 2011. The possible shapes of numerical ranges. arXiv preprint arXiv:1104.4587.
38. M. T. Chien, Y. H. Lin, 2000. On the area of numerical range. Ssoochow Journal of Mathematics, 26, Soochow University, pp.255–270.
39. Jeffrey Eldred, Leiba Rodman, Ilya Spitkovsky, 2012. Numerical ranges of companion matrices: flat portions on the boundary. Linear and Multilinear Algebra, 60, Taylor \& Francis, pp.1295–1311.
40. J Maroulas, P Psarrakos, 1996. Geometrical properties of numerical range of matrix polynomials. Computers \& Mathematics with Applications, 31, Pergamon, pp.41–47.
41. Moshe Goldberg, EG Straus, 1977. Elementary inclusion relations for generalized numerical ranges. Linear Algebra and Its Applications, 18, Elsevier, pp.1–24.
42. Ilya M Spitkovsky, Stephan Weis, 2015. Pre-images of extreme points of the numerical range, and applications. arXiv preprint arXiv:1509.05676.
43. Erin Militzer, Linda J Patton, Ilya M Spitkovsky, Ming-Cheng Tsai, 2017. Numerical Ranges of 4-by-4 Nilpotent Matrices: Flat Portions on the Boundary. Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, Springer, pp.561–591.

Joint numerical range

1. [44] 2. [45] 3. [46] 4. [47] 5. [48] 6. [49] 7. [50] 8. [51]

44. E. Gutkin, E.A. Jonckheere, M. Karow, 2004. Convexity of the joint numerical range: topological and differential geometric viewpoints. Linear algebra and its applications, 376, Elsevier, pp.143–171.
45. N. Krupnik, I. M. Spitkovsky, 2006. Sets of matrices with given joint numerical range. Linear algebra and its applications, 419, Elsevier, pp.569–585.
46. Konrad Szymański, 2017. Uncertainty relations and joint numerical ranges. arXiv preprint arXiv:1707.03464.
47. Konrad Szymański, Stephan Weis, Karol Życzkowski, 2017. Classification of joint numerical ranges of three hermitian matrices of size three. Linear Algebra and its Applications, Elsevier.
48. Chi-Kwong Li, Yiu-Tung Poon, Ya-Shu Wang, 2020. Joint numerical ranges and communtativity of matrices. arXiv preprint arXiv:2002.02768.
49. Daniel Plaumann, Rainer Sinn, Stephan Weis, 2019. Kippenhahn's Theorem for joint numerical ranges and quantum states. arXiv preprint arXiv:1907.04768.
50. Jakub Czartowski, Konrad Szymański, Bartłomiej Gardas, Yan V Fyodorov, Karol Życzkowski, 2019. Separability gap and large-deviation entanglement criterion. Physical Review A, 100, APS, pp.042326.
51. Konrad Jan Szymański, Karol Życzkowski, 2019. Geometric and algebraic origins of additive uncertainty relations. Journal of Physics A: Mathematical and Theoretical, IOP Publishing.

Restricted numerical ranges

1. [52] 2. [53] 3. [54] 4. [55] 5. [56] 6. [57] 7. [58] 8. [59]

52. A Abdollahi, 2006. The polynomial numerical hull of a matrix and algorithms for computing the numerical range. Applied mathematics and computation, 180, Elsevier, pp.635–640.
53. Hongkui Li, Xueting Liu, 2009. Elliptic Numerical Ranges of 4 x 4 Matrices. 2009 ETP International Conference on Future Computer and Communication, pp.190–193.
54. Z. Puchała, P. Gawron, J.A. Miszczak, Ł. Skowronek, M.D. Choi, K. Życzkowski, 2011. Product numerical range in a space with tensor product structure. Linear Algebra and its Applications, 434, Elsevier, pp.327–342.
55. J. W. Helton, I. M. Spitkovsky, 2011. The possible shapes of numerical ranges. arXiv:1104.4587, 1, pp.1-4.
56. Wai-Shun Cheung, Chi-Kwong Li, 2013. Elementary proofs for some results on the circular symmetry of the numerical range. Linear and Multilinear Algebra, 61, Taylor \& Francis, pp.596–602.
57. Jacek Jurkowski, Adam Rutkowski, D Chruściński, 2010. Local numerical range for a class of 2⊗ d Hermitian operators. Open Systems \& Information Dynamics, 17, World Scientific, pp.347–359.
58. Ryuusuke Koide, Hiroshi Nakazato, 2008. The q-numerical range of a certain 3$\times$ 3 matrix. International Mathematical Forum3, Citeseer, pp.1001–1010.
59. Chi-Kwong Li, 1998. q-Numerical ranges of normal and convex matrices. Linear and Multilinear Algebra, 43, Taylor \& Francis, pp.377–384.

Higher order numerical ranges

1. [60] 2. [61] 3. [62] 4. [63] 5. [64] 6. [65] 7. [66] 8. [67] 9. [68] 10. [69] 11. [70] 12. [71] 13. [72] 14. [73]

60. M. D. Choi, J. A. Holbrook, D. W, Kribs, K. Życzkowski, 2007. Higher-rank numerical ranges of unitary and normal matrices. Operators and Matrices, 1, pp.409–426.
61. M. D. Choi, M. Giesinger, J. A. Holbrook, D. W. Kribs, 2008. Geometry of higher-rank numerical ranges. Linear and Multilinear Algebra, 56, Taylor & Francis, pp.53–64.
62. H. J. Woerdeman, 2008. The higher rank numerical range is convex. Linear and Multilinear Algebra, 56, Taylor & Francis, pp.65–67.
63. C. K, Li, N. S. Sze, 2008. Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations. Proceedings of the American Mathematical Society, 136, pp.3013–3023.
64. C. K. Li, Y. T. Poon, N. S. Sze, 2009. Condition for the higher rank numerical range to be non-empty. Linear and Multilinear Algebra, 57, Taylor & Francis, pp.365–368.
65. Hwa-Long Gau, Chi-Kwong Li, Pei Yuan Wu, 2010. Higher-rank numerical ranges and dilations. Journal of Operator Theory, JSTOR, pp.181–189.
66. M. D. Choi, D. W. Kribs, K. Życzkowski, 2006. Higher-rank numerical ranges and compression problems. Linear algebra and its applications, 418, pp.828–839.
67. Mao-Ting Chien, Hiroshi Nakazato, 2011. The boundary of higher rank numerical ranges. Linear algebra and its applications, 435, Elsevier, pp.2971–2985.
68. Hwa-Long Gau, Pei Yuan Wu, 2013. Higher-rank numerical ranges and Kippenhahn polynomials. Linear Algebra and its Applications, 438, Elsevier, pp.3054–3061.
69. Hwa-Long Gau, Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze, 2011. Higher rank numerical ranges of normal matrices. SIAM Journal on Matrix Analysis and Applications, 32, SIAM, pp.23–43.
70. John Holbrook, Nishan Mudalige, Mike Newman, Rajesh Pereira, 2015. Bounds on polygons of higher rank numerical ranges. Linear Algebra and its Applications, 474, Elsevier, pp.230–242.
71. Martín Argerami, Saleh Mustafa, 2019. Higher rank numerical ranges of Jordan-like matrices. Linear and Multilinear Algebra, Taylor \& Francis, pp.1–20.
72. Chi-Kwong Li, Nam-Kiu Tsing, 1991. On the k th matrix numerical range. Linear and Multilinear Algebra, 28, Taylor and Francis, pp.229–239.
73. Mao-Ting Chien, Chi-Kwong Li, Hiroshi Nakazato, 2020. The diameter and width of higher rank numerical ranges. Linear and Multilinear Algebra, Taylor and Francis, pp.1–17.

Quaternion numerical ranges

1. [74] 2. [75] 3. [76] 4. [77]

74. G Najarbashi, S Ahadpour, MA Fasihi, Y Tavakoli, 2007. Geometry of a two-qubit state and intertwining quaternionic conformal mapping under local unitary transformations. Journal of Physics A: Mathematical and Theoretical, 40, IOP Publishing, pp.6481.
75. Luís Carvalho, Cristina Diogo, Sérgio Mendes, 2020. The star-center of the quaternionic numerical range. Linear Algebra and its Applications, Elsevier.
76. P Santhosh Kumar, 2019. A note on convexity of sections of quaternionic numerical range. Linear Algebra and its Applications, 572, Elsevier, pp.92–116.
77. Luis Carvalho, Cristina Diogo, Sérgio Mendes, 2019. On the convexity and circularity of the numerical range of nilpotent quaternionic matrices. arXiv preprint arXiv:1907.13438.

C numerical range

1. [78] 2. [79]

78. Yanfang Zhang, Xiaochun Fang, 2019. c-numerical range of operator products on B (H). arXiv preprint arXiv:1901.05245.
79. Gunther Dirr, Frederik vom Ende, 2018. The C-Numerical Range for Schatten-Class Operators. arXiv preprint arXiv:1808.06898.

$p$-th matricial range

1. [80] 2. [81] 3. [82] 4. [83]

80. Pan-Shun Lau, Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze, 2018. Convexity and star-shapedness of matricial range. Journal of Functional Analysis, 275, Elsevier, pp.2497–2515.
81. Robert C Thompson, 1987. Research problem the matrix numerical range. Taylor and Francis.
82. Chi-Kwong Li, Nam-Kiu Tsing, 1989. The numerical range of derivations. Linear Algebra and its Applications, 119, North-Holland, pp.97–119.
83. Wai-Fong Chuan, 1985. The unitary equivalence of compact operators. Glasgow Mathematical Journal, 26, Cambridge University Press, pp.145–149.

$(p,k)$ numerical range

1. [84] 2. [85] 3. [86]

84. Chi-Kwong Li, Bit-Shun Tam, Nam-Kiu Tsing, 1988. Linear operators preserving the (p, q)-numerical range. Linear Algebra and its Applications, 110, Elsevier, pp.75–89.
85. Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze, 2012. Generalized interlacing inequalities. Linear and Multilinear Algebra, 60, Taylor and Francis, pp.1245–1254.
86. Man-Duen Choi, Nathaniel Johnston, David W Kribs, 2009. The multiplicative domain in quantum error correction. Journal of Physics A: Mathematical and Theoretical, 42, IOP Publishing, pp.245303.

Essential numerical ranges

1. [87] 2. [88] 3. [89] 4. [90]

87. Sabine Bogli, Marco Marletta, 2019. Essential numerical ranges for linear operator pencils. arXiv preprint arXiv:1909.01301.
88. PA Fillmore, JG Stampfli, James P Williams, 1972. On the essential numerical range, the essential spectrum, and a problem of Halmos. Acta Sci. Math.(Szeged), 33, pp.179–192.
89. JG Stampfli, JP Williams, 1968. Growth conditions and the numerical range in a Banach algebra. Tohoku Mathematical Journal, Second Series, 20, Mathematical Institute, Tohoku University, pp.417–424.
90. Sabine Bogli, Marco Marletta, Christiane Tretter, 2020. The essential numerical range for unbounded linear operators. Journal of Functional Analysis, Elsevier, pp.108509.

Applications in quantum physics

1. [91] 2. [92] 3. [93]] 4. [94] 5. [95]

91. D. W. Kribs, A. Pasieka, M. Laforest, C. Ryan, M. P. da Silva, 2009. Research problems on numerical ranges in quantum computing. Linear and Multilinear Algebra, 57, Taylor & Francis, pp.491–502.
92. T. Schulte-Herbruggen, G. Dirr, U. Helmke, S. J. Glaser, 2008. The significance of the C-numerical range and the local C-numerical range in quantum control and quantum information. Linear and Multilinear Algebra, 56, Taylor & Francis, pp.3–26.
93. Piotr Gawron, Zbigniew Puchała, Jarosław Adam Miszczak, Łukasz Skowronek, Karol Życzkowski, 2010. Restricted numerical range: a versatile tool in the theory of quantum information. Journal of mathematical physics, 51, American Institute of Physics, pp.102204.
94. K. Życzkowski, M.-D. Choi, C. Dunkl, J. Holbrook, P. Gawron, J. A. Miszczak, Z. Puchala, Ł. Skowronek, 2009. Generalized numerical range as a versatile tool to study quantum entanglement. Oberwolfach Report, 59, pp.34-37.
95. Ningping Cao, David W Kribs, Chi-Kwong Li, Mike I Nelson, Yiu-Tung Poon, Bei Zeng, 2020. Higher rank matricial ranges and hybrid quantum error correction. Linear and Multilinear Algebra, Taylor \& Francis, pp.1–13.

Numerical shadow

General

1. [96] 2. [97] 3. [98] 4. [99] 5. [100]

96. C.F. Dunkl, P. Gawron, J.A. Holbrook, J.A. Miszczak, Z. Puchała, K. Życzkowski, 2011. Numerical shadow and geometry of quantum states. Journal of Physics A: Mathematical and Theoretical, 44, IOP Publishing, pp.335301.
97. C.F. Dunkl, P. Gawron, J.A. Holbrook, Z. Puchała, K. Zyczkowski, 2011. Numerical shadows: measures and densities on the numerical range. Linear Algebra and its Applications, 434, North-Holland, pp.2042–2080.
98. Ingemar Bengtsson, Stephan Weis, Karol Życzkowski, 2013. Geometry of the set of mixed quantum states: An apophatic approach. Geometric Methods in Physics, Springer, pp.175–197.
99. Eugene Gutkin, Karol Życzkowski, 2013. Joint numerical ranges, quantum maps, and joint numerical shadows. Linear Algebra and its Applications, 438, Elsevier, pp.2394–2404.
100. Thierry Gallay, Denis Serre, 2012. Numerical measure of a complex matrix. Communications on Pure and Applied Mathematics, 65, Wiley Online Library, pp.287–336.

Restricted numerical shadow

1. [101] 2. [102]

101. Z. Puchała, J.A. Miszczak, P. Gawron, C.F. Dunkl, J.A. Holbrook, K. Życzkowski, 2012. Restricted numerical shadow and geometry of quantum entanglement. Journal of Physics A: Mathematical and Theoretical, 45, pp.415309.
102. Charles F Dunkl, Piotr Gawron, Łukasz Pawela, Zbigniew Puchała, Karol Życzkowski, 2015. Real numerical shadow and generalized B-splines. Linear Algebra and its Applications, 479, Elsevier, pp.12–51.

Quantum canonical ensemble

1. [103] 2. [104] 3. [105] 4. [106] 5. [107]

103. Dorje C Brody, Lane P Hughston, 1998. The quantum canonical ensemble. Journal of Mathematical Physics, 39, American Institute of Physics, url=https://aip.scitation.org/doi/abs/10.1063/1.532661, pp.6502–6508.
104. D. C. Brody, D. W. Hook, L. P. Hughston, 2005. Microcanonical distributions for quantum systems. arxiv, 1, pp.1-8.
105. Dorje C Brody, Daniel W Hook, Lane P Hughston, 2007. On quantum microcanonical equilibrium. Journal of Physics: Conference Series, pp.012025.
106. Dorje C Brody, Daniel W Hook, Lane P Hughston, 2007. Quantum phase transitions without thermodynamic limits. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463, The Royal Society London, pp.2021–2030.
107. L Campos Venuti, Paolo Zanardi, 2013. Probability density of quantum expectation values. Physics Letters A, 377, Elsevier, pp.1854–1861.
literature.txt · Last modified: 2020/06/19 08:32 by rkukulski

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