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New Weyl-Type Inequalities by Multiplicative Injective and Surjective s-Numbers of Operators in Reflexive Banach Spaces

Year 2022, Volume: 12 Issue: 1, 40 - 55, 30.06.2022
https://doi.org/10.37094/adyujsci.1001762

Abstract

In this work, two problems are investigated. In general, Weyl-type inequalities of operators in complex reflexive Banach spaces are discussed. First, we obtained the Weyl-type inequalities using arbitrary multiplicative surjective and injective 𝑠-numbers that are dual of each other. Second, we introduced the Weyl-type inequalities by multiplicative injective and surjective 𝑠- numbers under certain conditions for 𝑆 and 𝑆^' operators in complex reflexive Banach space. So, new Weyl-type inequalities are investigated for both dual 𝑠-number sequences and dual operators.

References

  • [1] Pietsch, A., 𝑠-number of operators in Banach spaces, Studia Math., 51, 201-233, 1974.
  • [2] Weyl, H., Inequalities between two kinds of eigenvalues of a linear transformation, Proceedings of National Academy of Sciences USA, 35, 408-411, 1949.
  • [3] Pietsch, A., Weyl numbers and eigenvalues of operators in Banach spaces, Mathematische Annalen, 247, 149-168, 1980.
  • [4] Carl, B., Hincichs, A., Optimal Weyl-type inequalities for operators in Banach spaces, Positivity, 11, 41-55, 2007.
  • [5] Pietsch, A., Eigenvalues and 𝑠-numbers, Cambridge University Press, 1987.
  • [6] König, H., Eigenvalues of compact operators with applications to integral operators, Linear Algebra and its Applications, 84, 111-122, 1986.
  • [7] König, H., Eigenvalues of operators and aplication, Handbook of Geometry of Banach Spaces, North-Holland, Amsterdam, 1, 941-974, 2001.
  • [8] Carl, B., On a Weyl inequalities of operators in Banach spaces, Proceedings of the American Mathematical Society, 137, 155-159, 2009.
  • [9] Carl, B., Hincichs, A., On s-numbers and Weyl inequalities of operators in Banach spaces, The Bulletin of the London Mathematical Society, 41, 2, 332-340, 2009.
  • [10] Kochubei, A.N., Symmetric Operators and Nonclassical Specktral Problems, Matematicheskie Zametki, 25, 425-434, 1979.
  • [11] Barramov, E., Öztürk Mert, R., Ismailov, Z.I., Selfadjoint Extensions of a Singular Differential Operators, Journal of Mathematical Chemistry, 50, 1100-1110, 2012.
  • [12] Ismailov, Z.I, Cona L., Cevik, E.O., Guler, B.O., Weyl Numbers of Diagonal Matrices, AIP Conference Proceedings, Vol. 1611, 296-299, 2014.
  • [13] Ismailov, Z.I, Cona L., Cevik, E.O., Gelfand Numbers of Diagonal Matrices, Hacettepe Journal of Mathematics and Statistics, 44(1), 75-81, 2015.
  • [14] Pietsch, A., Operator ideals, North Holland, Amsterdam, New York, Oxford, 1980.
  • [15] Pietsch, A., History of Banach spaces and linear operators, Birkhäuser, 2007.
  • [16] Timoshenkox, A., Theory of elastic stability, Mc Grow-Hill, New York, 1961.
  • [17] Ipek Al, P., Ismailov, Z.I., Lorentz-Schatten characteristic of compact inverses of first order normal differential operators, The Mediterranean International Conference of Pure Applied Mathematics and Related Areas (MICOPAM 2018), 200-203, 2018.
  • [18] Ipek Al, P., Ismailov, Z.I., Singular numbers of lower triangular one-band block operator matrices, International Conference on Mathematics “An Istanbul Meeting for World Mathematicians” Minisymposium on Approximation Theory Minisymposium on Math Education (ICOM 2018), 184-189, 2018.
  • [19] Öztürk Mert, R., Ipek Al, P., Ismailov, Z.I, Lorentz-Schatten property of the inverses of second order differential operators with Dirichlet conditions, Current Academic Studies in Natural Science and Mathematics Sciences, Mehmet Ali Kandemir, Fatma Erdoğan, Editor, IVPE, ss.155-161, 2019.
  • [20] Ipek Al, P., Ismailov, Z.I., Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices, Communications Faculty of Science University of Ankara Series A1 Mathematics and Statistics, 68(2), 1852-1866, 2019.
  • [21] Ipek Al, P., Ismailov, Z.I., Compact inverses of first order normal differential operators with Lorentz-Schatten properties, 3. International Conference on Mathematics: An Istanbul Meeting for World Mathematicians (ICOM 2019), 359-363, 2019.
  • [22] Ipek Al, P., Ismailov, Z.I., Schatten-von Neumann characteristic of tensor product operators, Filomat, 34(10), 2020.
  • [23] Ipek Al, P., Lorentz-Schatten classes of direct sum of operators, Hacettepe Journal of Mathematics and Statistics, 49(2), 835-842, 2020.
  • [24] Ipek Al, P., Lorentz-Marcinkiewicz property of direct sum of operators, Filomat, 34(2), 391-398, 2020.

Yansımalı Banach Uzaylarda Operatörlerin Çarpımsal İnjektiv ve Surjektiv s-Sayıları ile Yeni Weyl-Tipi Eşitsizlikleri

Year 2022, Volume: 12 Issue: 1, 40 - 55, 30.06.2022
https://doi.org/10.37094/adyujsci.1001762

Abstract

Bu çalışmada iki problem incelenmiştir. Genel olarak, kompleks yansımalı Banach uzaylarında operatörlerin Weyl-tipi eşitsizlikleri üzerinde durulmuştur. İlk olarak, birbirinin duali olan keyfi çarpımsal surjektif ve injektif 𝑠-sayılarını kullanarak Weyl-tipi eşitsizlikler elde edilmiştir. İkinci olarak, kompleks yansımalı Banach uzayındaki 𝑆 ve 𝑆$ operatörleri için belirli koşullar altında çarpımsal injektif ve surjektif 𝑠-sayıları ile Weyl-tipi eşitsizlikler ifade edilmiştir. Böylece hem dual 𝑠-sayı dizileri hem de dual operatörler için yeni Weyl-tipi eşitsizlikleri araştırılmıştır.

References

  • [1] Pietsch, A., 𝑠-number of operators in Banach spaces, Studia Math., 51, 201-233, 1974.
  • [2] Weyl, H., Inequalities between two kinds of eigenvalues of a linear transformation, Proceedings of National Academy of Sciences USA, 35, 408-411, 1949.
  • [3] Pietsch, A., Weyl numbers and eigenvalues of operators in Banach spaces, Mathematische Annalen, 247, 149-168, 1980.
  • [4] Carl, B., Hincichs, A., Optimal Weyl-type inequalities for operators in Banach spaces, Positivity, 11, 41-55, 2007.
  • [5] Pietsch, A., Eigenvalues and 𝑠-numbers, Cambridge University Press, 1987.
  • [6] König, H., Eigenvalues of compact operators with applications to integral operators, Linear Algebra and its Applications, 84, 111-122, 1986.
  • [7] König, H., Eigenvalues of operators and aplication, Handbook of Geometry of Banach Spaces, North-Holland, Amsterdam, 1, 941-974, 2001.
  • [8] Carl, B., On a Weyl inequalities of operators in Banach spaces, Proceedings of the American Mathematical Society, 137, 155-159, 2009.
  • [9] Carl, B., Hincichs, A., On s-numbers and Weyl inequalities of operators in Banach spaces, The Bulletin of the London Mathematical Society, 41, 2, 332-340, 2009.
  • [10] Kochubei, A.N., Symmetric Operators and Nonclassical Specktral Problems, Matematicheskie Zametki, 25, 425-434, 1979.
  • [11] Barramov, E., Öztürk Mert, R., Ismailov, Z.I., Selfadjoint Extensions of a Singular Differential Operators, Journal of Mathematical Chemistry, 50, 1100-1110, 2012.
  • [12] Ismailov, Z.I, Cona L., Cevik, E.O., Guler, B.O., Weyl Numbers of Diagonal Matrices, AIP Conference Proceedings, Vol. 1611, 296-299, 2014.
  • [13] Ismailov, Z.I, Cona L., Cevik, E.O., Gelfand Numbers of Diagonal Matrices, Hacettepe Journal of Mathematics and Statistics, 44(1), 75-81, 2015.
  • [14] Pietsch, A., Operator ideals, North Holland, Amsterdam, New York, Oxford, 1980.
  • [15] Pietsch, A., History of Banach spaces and linear operators, Birkhäuser, 2007.
  • [16] Timoshenkox, A., Theory of elastic stability, Mc Grow-Hill, New York, 1961.
  • [17] Ipek Al, P., Ismailov, Z.I., Lorentz-Schatten characteristic of compact inverses of first order normal differential operators, The Mediterranean International Conference of Pure Applied Mathematics and Related Areas (MICOPAM 2018), 200-203, 2018.
  • [18] Ipek Al, P., Ismailov, Z.I., Singular numbers of lower triangular one-band block operator matrices, International Conference on Mathematics “An Istanbul Meeting for World Mathematicians” Minisymposium on Approximation Theory Minisymposium on Math Education (ICOM 2018), 184-189, 2018.
  • [19] Öztürk Mert, R., Ipek Al, P., Ismailov, Z.I, Lorentz-Schatten property of the inverses of second order differential operators with Dirichlet conditions, Current Academic Studies in Natural Science and Mathematics Sciences, Mehmet Ali Kandemir, Fatma Erdoğan, Editor, IVPE, ss.155-161, 2019.
  • [20] Ipek Al, P., Ismailov, Z.I., Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices, Communications Faculty of Science University of Ankara Series A1 Mathematics and Statistics, 68(2), 1852-1866, 2019.
  • [21] Ipek Al, P., Ismailov, Z.I., Compact inverses of first order normal differential operators with Lorentz-Schatten properties, 3. International Conference on Mathematics: An Istanbul Meeting for World Mathematicians (ICOM 2019), 359-363, 2019.
  • [22] Ipek Al, P., Ismailov, Z.I., Schatten-von Neumann characteristic of tensor product operators, Filomat, 34(10), 2020.
  • [23] Ipek Al, P., Lorentz-Schatten classes of direct sum of operators, Hacettepe Journal of Mathematics and Statistics, 49(2), 835-842, 2020.
  • [24] Ipek Al, P., Lorentz-Marcinkiewicz property of direct sum of operators, Filomat, 34(2), 391-398, 2020.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Lale Cona 0000-0002-2744-1960

Publication Date June 30, 2022
Submission Date September 27, 2021
Acceptance Date April 5, 2022
Published in Issue Year 2022 Volume: 12 Issue: 1

Cite

APA Cona, L. (2022). New Weyl-Type Inequalities by Multiplicative Injective and Surjective s-Numbers of Operators in Reflexive Banach Spaces. Adıyaman University Journal of Science, 12(1), 40-55. https://doi.org/10.37094/adyujsci.1001762
AMA Cona L. New Weyl-Type Inequalities by Multiplicative Injective and Surjective s-Numbers of Operators in Reflexive Banach Spaces. ADYU J SCI. June 2022;12(1):40-55. doi:10.37094/adyujsci.1001762
Chicago Cona, Lale. “New Weyl-Type Inequalities by Multiplicative Injective and Surjective S-Numbers of Operators in Reflexive Banach Spaces”. Adıyaman University Journal of Science 12, no. 1 (June 2022): 40-55. https://doi.org/10.37094/adyujsci.1001762.
EndNote Cona L (June 1, 2022) New Weyl-Type Inequalities by Multiplicative Injective and Surjective s-Numbers of Operators in Reflexive Banach Spaces. Adıyaman University Journal of Science 12 1 40–55.
IEEE L. Cona, “New Weyl-Type Inequalities by Multiplicative Injective and Surjective s-Numbers of Operators in Reflexive Banach Spaces”, ADYU J SCI, vol. 12, no. 1, pp. 40–55, 2022, doi: 10.37094/adyujsci.1001762.
ISNAD Cona, Lale. “New Weyl-Type Inequalities by Multiplicative Injective and Surjective S-Numbers of Operators in Reflexive Banach Spaces”. Adıyaman University Journal of Science 12/1 (June 2022), 40-55. https://doi.org/10.37094/adyujsci.1001762.
JAMA Cona L. New Weyl-Type Inequalities by Multiplicative Injective and Surjective s-Numbers of Operators in Reflexive Banach Spaces. ADYU J SCI. 2022;12:40–55.
MLA Cona, Lale. “New Weyl-Type Inequalities by Multiplicative Injective and Surjective S-Numbers of Operators in Reflexive Banach Spaces”. Adıyaman University Journal of Science, vol. 12, no. 1, 2022, pp. 40-55, doi:10.37094/adyujsci.1001762.
Vancouver Cona L. New Weyl-Type Inequalities by Multiplicative Injective and Surjective s-Numbers of Operators in Reflexive Banach Spaces. ADYU J SCI. 2022;12(1):40-55.

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