Full-time Faculty
Photo of Hwa-Long Gau
Professor

Hwa-Long Gau

Research areas

  • Operator theory
  • Matrix analysis

Education

Ph.D., National Chiao Tung University

Experience

2013/01 ~ present Linear and Multilinear Algebra, Editorial board member
2011/08 ~ present Professor, National Central University
2005/08 ~ 2011/07 Associate Professor, National Central University
2001/08 ~ 2005/07 Assistant Professor, National Central University
2000/05 ~ 2001/07 Postdoctor, National Sun Yat-sen University

Honors

國立中央大學109學年度「理學院優良教師」

國立中央大學108學年度「理學院優良教師」

國立中央大學100學年度「學術研究傑出獎」
國立中央大學99學年度「理學院優良教師」
國立中央大學98學年度「理學院優良教師」
國立中央大學93學年度「理學院優良教師」
 

Publications

(A) Books:

  1. Pei Yuan Wu and Hwa-Long Gau, Numerical Ranges of Hilbert Space Operators, Cambridge University Press, Cambridge, 2021.  (Corrections to the book)

 

(B) Papers:

(A) 期刊論文

  1. Hwa-Long Gau and Pei Yuan Wu*, A brief survey on ρ-contractions and unitary ρ-dilations,  Acta Sci. Math. (Szeged) (2026), to appear.
  2. Hwa-Long Gau* and Pei Yuan Wu, Direct decomposition of ρ-contraction and the unitary ρ-dilation, Proc. Amer. Math. Soc. 154 (2026), 1167--1180.
  3. Hwa-Long Gau, Chi-Kwong Li and Kuo-Zhong Wang*, Matrices with all diagonal entries lying on the boundary of the numerical range, Linear Algebra Appl. 733 (2026), 1--25.
  4. Hwa-Long Gau* and Pei Yuan Wu, ρ-Contraction and its consequences, Linear Algebra Appl. 729 (2026), 246--255.
  5. Hwa-Long Gau and Kuo-Zhong Wang*, Constants related to powers of ρ-contractions, J. Math. Anal. Appl. 547 (2025), no. 1, Paper No. 129345.
  6. Hwa-Long Gau* and Pei Yuan Wu, Compressions and direct sums, Adv. Oper. Theory 9 (2024), no. 1, Paper No. 5, 14 pp.
  7. Hwa-Long Gau*, Proof of a conjecture on numerical ranges of weighted cyclic matrices, Linear Algebra Appl. 682 (2024), 295--308.
  8. Pei Yuan Wu* and Hwa-Long Gau, Numerical ranges of operators and matrices, Math. Online First Collect., Springer, Cham, 2023, 413--439.
  9. Hwa-Long Gau*, Chi-Kwong Li, Kuo-Zhong Wang, and Pei Yuan Wu, Extremality of bounds for numerical radii of Foguel operators, J. Math. Anal. Appl. 528 (2023), no. 1, Paper No. 127484, 18 pp.
  10. Pei Yuan Wu* and Hwa-Long Gau, Which set is the numerical range of an operator? Acta Sci. Math. (Szeged) 88 (2022), no. 1-2, 527--545.
  11. Hwa-Long Gau* and Pei Yuan Wu, Numerical ranges of Hankel matrices, Linear Algebra Appl. 650 (2022), 60--74.
  12. Hwa-Long Gau*, Kuo-Zhong Wang and Pei Yuan Wu, Numerical range of the Foguel-Halmos operator, Studia Math. 263 (2022), no. 3, 267--291.
  13. Hwa-Long Gau*, Kuo-Zhong Wang and Pei Yuan Wu, Numerical ranges of Foguel operators, Linear Algebra Appl. 610 (2021), 766--784.
  14. Hwa-Long Gau and Kuo-Zhong Wang*, Matrix powers with circular numerical range, Linear Algebra Appl. 603 (2020), 190--211.
  15. Hwa-Long Gau*, Kuo-Zhong Wang and Pei Yuan Wu, Constant norms and numerical radii of matrix powers, Operators and Matrices, 13 (2019), 1035--1054.
  16. Hwa-Long Gau*, Kuo-Zhong Wang and Pei Yuan Wu, Equality of numerical ranges of matrix powers, Linear Algebra Appl. 578 (2019), 95--110.
  17. Hwa-Long Gau* and Yueh-Hua Lu, Extremality of numerical radii of tensor products of matrices, Linear Algebra Appl. 565 (2019), 82--98.
  18. Hwa-Long Gau* and Pei Yuan Wu, Equality of three numerical radius inequalities, Linear Algebra Appl. 554 (2018), 51--67.
  19. Hwa-Long Gau* and Pei Yuan Wu, Lower bounds for the numerical radius, Operators and Matrices, 11 (2017), no. 4, 999--1014.
  20. Hwa-Long Gau* and Pei Yuan Wu, Operators with real parts at least -1/2, Linear Multilinear Algebra, 65 (2017), no. 10, 1988--1999.
  21. Hwa-Long Gau and Kuo-Zhong Wang*, Extremality of numerical radii of matrix commutators and Jordan products, Linear Algebra Appl. 533 (2017), 258--281.
  22. Hwa-Long Gau* and Pei Yuan Wu, Numerical ranges of products of two positive contractions, J. Math. Anal. Appl. 455 (2017), 939--946.
  23. Hwa-Long Gau* and Pei Yuan Wu, Zero-dilation index of Sn-matrix and companion matrix, Electronic Journal of Linear Algebra, 31 (2016), 666-678.
  24. Hwa-Long Gau, Kuo-Zhong Wang* and Pei Yuan Wu, Numerical ranges of row stochastic matrices, Linear Algebra Appl. 506 (2016), 478--505.
  25. Hwa-Long Gau* and Pei Yuan Wu, Numerical radius of Hadamard product of matrices, Linear Algebra Appl. 504 (2016), 292--308.
  26. Hwa-Long Gau* and Pei Yuan Wu, Extremality of numerical radii of matrix products, Linear Algebra Appl. 501 (2016), 17--36.
  27. Hwa-Long Gau*, Kuo-Zhong Wang and Pei Yuan Wu, Crawford numbers of companion matrices, Operators and Matrices, 10 (2016), no. 4, 863–880.
  28. Mao-Ting Chien, Hwa-Long Gau*, Chi-Kwong Li, Ming-Cheng Tsai and Kuo-Zhong Wang Product of operators and numerical range, Linear Multilinear Algebra, 64 (2016), no. 1, 58–67.
  29. Hwa-Long Gau*, Kuo-Zhong Wang and Pei Yuan Wu, Circular numerical ranges of partial isometries, Linear Multilinear Algebra, 64 (2016), no. 1, 14–35.
  30. Hwa-Long Gau and Pei Yuan Wu*, Upper and lower bounds for numerical radii of block shifts, Bull. Iranian Math. Soc., 41 (2015), no. 7, 15–27.
  31. Hwa-Long Gau*, Kuo-Zhong Wang and Pei Yuan Wu, Numerical radii for tensor products of matrices, Linear Multilinear Algebra 63 (2015), 1916--1936.
  32. Hwa-Long Gau* and Pei Yuan Wu, Power partial isometry index and ascent of a finite matrix, Linear Algebra Appl. 459 (2014), 136–144.
  33. Hwa-Long Gau and Pei Yuan Wu*, Excursions in numerical ranges, Bull. Inst. Math. Acad. Sin. (N. S.), 9 (2014), No. 3, 351--370.
  34. Hwa-Long Gau, Kuo-Zhong Wang and Pei Yuan Wu*, Numerical radii for tensor products of operators, Integral Equations and Operator Theory, 78 (2014), 375--382.
  35. Chi-Tung Chang*, Hwa-Long Gau and Kuo-Zhong Wang, Equality of higher-rank numerical ranges of matrices, Linear Multilinear Algebra 62 (2014), 626--638.
  36. Ming-Cheng Tsai*, Hwa-Long Gau and Han-Chun Wang, Refinements of numerical ranges of weighted shift matrices, Linear Multilinear Algebra 62 (2014), 568--578.
  37. Hwa-Long Gau* and Pei Yuan Wu, Structures and numerical ranges of power partial isometries, Linear Algebra Appl. 440 (2014), 325–341.
  38. Hwa-Long Gau*, Kuo-Zhong Wang and Pei Yuan Wu, Zero-dilation index of a finite matrix, Linear Algebra Appl. 440 (2014), 111–124.
  39. Hwa-Long Gau and Pei Yuan Wu*, Zero-dilation indices of KMS matrices, Ann. Funct. Anal. 5 (2014), no. 1, 30--35.
  40. Hwa-Long Gau and Pei Yuan Wu*, Numerical ranges of KMS matrices, Acta Sci. Math. (Szeged) 79 (2013), 583–610.
  41. Cheng-Chun Chang, Hwa-Long Gau, Ya-Shu Wang*, Szu-Chieh Wu and Ya-Ting Yeh, Matrices with defect index one, Operators and Matrices 7 (2013), no. 4, 865–885.
  42. Hwa-Long Gau and Pei Yuan Wu*, Numerical ranges and compressions of Sn-matrices, Operators and Matrices 7 (2013), no. 2, 465–476.
  43. Hwa-Long Gau* and Pei Yuan Wu, Higher-rank numerical ranges and Kippenhahn polynomials, Linear Algebra Appl. 438 (2013), 3054–3061.
  44. Chi-Tung Chang*, Hwa-Long Gau, Kuo-Zhong Wang and Pei Yuan Wu, Numerical ranges and Gersgorin discs, Linear Algebra Appl. 438 (2013), 1170–1192.
  45. Hwa-Long Gau*, Ming-Cheng Tsai and Han-Chun Wang, Weighted shift matrices: unitary equivalence, reducibility and numerical ranges, Linear Algebra Appl. 438 (2013), 498–513.
  46. Hwa-Long Gau* and Pei Yuan Wu, Noncircular elliptic discs as numerical ranges of nilpotent operators, Linear Multilinear Algebra 60 (2012), 1225-1233.
  47. Hwa-Long Gau, Chi-Kwong Li, Yiu-Tung Poon and Nung-Sing Sze*, Higher rank numerical ranges of normal matrices, SIAM. J. Matrix Anal. Appl. 32 (2011), Issue 1, 23-43.
  48. Kuo-Zhong Wang, Pei Yuan Wu* and Hwa-Long Gau, Crawford numbers of powers of a matrix, Linear Algebra Appl. 433 (2010), 2243--2254.
  49. Hwa-Long Gau* and Pei Yuan Wu, Defect indices of powers of a contraction, Linear Algebra Appl. 432 (2010), 2824--2833.
  50. Hwa-Long Gau* and Pei Yuan Wu, Unitary part of a contraction, J. Math. Anal. Appl. 366 (2010), 700--705.    .
  51. Hwa-Long Gau*, Numerical ranges of reducible companion matrices, Linear Algebra Appl. 432 (2010), 1310--1321.
  52. Hwa-Long Gau*, Chi-Kwong Li and Pei Yuan Wu, Higher-rank numerical ranges and dilations, Journal of Operator Theory. 63:1 (2010),181--189.
  53. Hwa-Long Gau* and Pei Yuan Wu, Inner functions of numerical contractions, Linear Algebra Appl. 430 (2009), 2182--2191.
  54. Pei Yuan Wu*, Hwa-Long Gau, and Ming Cheng Tsai, Numerical radius inequality for C0 contractions, Linear Algebra Appl. 430 (2009), 1509--1516.
  55. Hwa-Long Gau* and Pei Yuan Wu, Numerical ranges of nilpotent operators, Linear Algebra Appl. 429 (2008), 716--726.
  56. Hwa-Long Gau, Chih-Jen Wang and Ngai-Ching Wong*, Invertibility and Fredholmness of linear combinations of quadratic, k-potent and nilpotent operators, Opreators and Matrices, 2 (2008), no. 2, 193--199.
  57. Hwa-Long Gau, Chin-Ying Huang and Pei Yuan Wu*, Numerical radius inequalities for square-zero and idempotent operators, Opreators and Matrices, 2 (2008), no. 1, 137--141.
  58. Hwa-Long Gau* and Pei Yuan Wu, Line segments and elliptic arcs on the boundary of a numerical range, Linear Multilinear Algebra 56 (2008), no. 1-2, 131--142.
  59. Hwa-Long Gau and Pei Yuan Wu*, Fredholmness of linear combinations of two idempotents, Integral Equations and Operator Theory, 59 (2007), 579--583.
  60. Hwa-Long Gau* and Chi-Kwong Li, C*-isomorphisms, Jordan isomorphisms, and numerical range preserving maps, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2907—2914.
  61. Hwa-Long Gau and Pei Yuan Wu*, Numerical ranges of companion matrices, Linear Algebra Appl. 421 (2007), 202--218.
  62. Hwa-Long Gau*, Elliptic numerical ranges of 4x4 matrices, Taiwanese J. Math, vol. 10, no. 1 (2006), 117--128.
  63. Hwa-Long Gau and Pei Yuan Wu*, Anderson's theorem for compact operators, Proc. Amer. Math. Soc., vol. 134, no. 11 (2006), 3159--3162.
  64. Hwa-Long Gau* and Pei Yuan Wu, Numerical range of a normal compression. II, Linear Algebra Appl. 390 (2004), 121--136.
  65. Hwa-Long Gau* and Pei Yuan Wu, Numerical range of a normal compression, Linear Multilinear Algebra, vol. 52, no. 3-4 (2004), 195--201.
  66. Hwa-Long Gau and Pei Yuan Wu*, Companion matrices: reducibility, numerical range and similarity to contractions, Linear Algebra Appl., 383 (2004), 127-142.
  67. Hwa-Long Gau and Pei Yuan Wu*, Numerical range circumscribed by two polygons, Linear Algebra Appl., 382 (2004), 155--170.
  68. Hwa-Long Gau and Pei Yuan Wu*, Finite Blaschke products of contractions, Linear Algebra Appl. 368 (2003), 359--370.
  69. Hwa-Long Gau and Pei Yuan Wu*, Numerical range and Poncelet property, Taiwanese J. Math. 7 (2003), no. 2, 173--193.
  70. Hwa-Long Gau and Pei Yuan Wu*, Condition for the numerical range to contain an elliptic disc, Linear Algebra Appl. 364 (2003), 213--222.
  71. Hwa-Long Gau*, Jyh-Shyang Jeang and Ngai-Ching Wong, Biseparating linear maps between continuous vector-valued function spaces, J. Aust. Math. Soc. 74 (2003), no. 1, 101--109.
  72. Hwa-Long Gau, Jyh-Shyang Jeang and Ngai-Ching Wong*, An algebraic approach to the Banach-Stone theorem for separating linear bijections, Taiwanese J. Math. 6 (2002), no. 3, 399--403.
  73. Hwa-Long Gau and Pei Yuan Wu*, Lucas' theorem refined, Linear and Multilinear Algebra 45 (1999), no. 4, 359--373.
  74. Hwa-Long Gau and Pei Yuan Wu*, Dilation to inflations of S(φ), Linear and Multilinear Algebra 45 (1998), no. 2-3, 109--123.
  75. Hwa-Long Gau and Pei Yuan Wu*, Numerical range of S(φ), Linear and Multilinear Algebra 45 (1998), no. 1, 49--73.
  76. Hwa-Long Gau and Ngai-Ching Wong*, Some converses of the strong separation theorem, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2443--2449.