Ajay Kumar
Professor,
Department of Mathematics, Faculty of Mathematical Sciences Building,
University of Delhi, Delhi 110 007, India

Phone :91-11-27666658
E-mail: akumar@maths.du.ac.in
Educational Qualifications

  1. Ph.D.,1980, University of Delhi.
  2. M.Sc. Mathematics,1976, University of Delhi.
  3. B.Sc.(Hons.) Mathematics,1974, University of Delhi.
  4. Proficiency in German Language, Studied at Goethe Institute, Rothenberg ob der Tauber, Germany,1981.
  5. Proficiency in French Language, Studied at University of Strasbourg, France,1987.


Positions held
  1. Dean Research (PS & MS), University of Delhi, July 4, 2012 - December 2, 2016.
  2. Head, Department of Mathematics, University of Delhi, Dec 21, 2012- Dec 20, 2015.
  3. Dean, Faculty of Mathematical Sciences, University of Delhi, Dec 21, 2012- August 2, 2013.
  4. Dean Research, University of Delhi, Jan. 24, 2011- July 3, 2012.
  5. Professor, Department of Mathematics, University of Delhi, Delhi, India since Sept.,2004.
  6. Postdoctoral fellow / Guest Professor in several Universities in Germany, France, U.K.
  7. Reader, Department of Mathematics, Rajdhani College (University of Delhi) from 1987-2004.
  8. Lecturer, Department of Mathematics, Rajdhani College (University of Delhi) from 1977-1987.
Fellowships and Research grants awarded
FOREIGN FELLOWSHIPS
  1. DAAD (German Academic Exchange Service) Post-doctoral fellow at University of Paderborn,Germany, 1981-1983.
  2. C.I.E.S.(French Govt. Fellowship) Post-doctoral fellow at University of Strasbourg,France, 1987-88.
  3. Post-doctoral Fellow (Wissenschaftlicher Mitarbeiter) at Fachbereich Mathematik, University of Paderborn, Germany, 1989.
  4. DAAD (German Academic Exchange Service) Re-invitation fellowship at Freie Universitat,Berlin,Germany, 1992.
  5. Post-doctoral Fellow of Freie Universitat Fachbereich Mathematik, Freie Universitat, Berlin, Germany, 1992.
  6. Commonwealth Academic Staff Fellowship at University of Edinburgh,U.K.1995-96
  7. DAAD (German Academic Exchange Service) Re-invitation fellowship at Fachbereich Mathematik, University of Paderborn,Germany, 1999.
  8. Royal Society London fellowship, University of Edinburgh, U.K., 2000.
  9. DAAD (German Academic Exchange Service) Re-invitation fellowship at Freie Universitat,Berlin,Germany, 2003.
  10. DFG (German Research Foundation), Freie Universitat,Berlin,Germany, 2004.
  11. DFG (German Research Foundation), Freie Universitat,Berlin,Germany, 2008.
  12. JSPS (Japan Society for Promotion of Science), Gunma University, Japan, 2012.

Research Experience: 1976-todate
  1. Specializations: Functional Analysis, Banach Algebras, Operator Theory, Operator Spaces, Harmonic Analysis, Representation theory of locally compact groups and hypergroups, Complex analytic methods in partial differential equations.
  2. Research Publications in journal of International repute namely Transactions of American Mathematical Society; Journal of Functional Analysis; Proceedings of American Mathematical Society; Pacific Journal of Mathematics; Mathematische Zeitschrifte; Complex Variables, and Elliptic Equations; Proceedings of Edinburgh Mathematical Society; Mathematical Proceedings of Cambridge Philosophical Society; Forum Mathematicum, Applicable Analysis, Journal of Australian Math.Soc.; Journal of Geometric Analysis; Analysis etc. (see List of Publications for details).
Math Geneology:
G. H. (Godfrey Harold) Hardy—>Frank Smithies—>Ajit Iqbal Singh—>Ajay Kumar


Collaborations with Foreign Universities:
  • Begehr,Heinrich G. W.,Freie Universitat Berlin, Germany.
  • Bhatta, Chet Raj, Universal College, Kathmandu, Nepal.
  • Echterhoff, Siegfried, Universitat Munster, Germany.
  • Gebuhrer, Marc-Oliver, I.R.M.A., Strasbourg, France.
  • Hauenschild, Wilfried, Universitat Paderborn, Germany.
  • Kaniuth, Eberhard, Universitat Pederborn, Germany.
  • Schmersau, Dieter, Freie Universitat Berlin, Germany.
  • Sinclair, Allan M., University of Edinburgh, U.K.
  • Vanegas, Carmen Judith, Univ. Simon Bolivar, Venezuela.
  • Takashi Itoh, Gunma University, Japan.

Publications

  1. Ajit Kaur Chilana and Ajay Kumar, (1979) Spectral synthesis in Segal algebras on hypergroups. Pacific Journal of Mathematics 80,59-76(USA) Mathematical Reviews 80g:430011 Zentralblatt fur Mathematik 371.43012(402.43004)
  2. Ajit Kaur Chilana and Ajay Kumar, (1979) Ultra strong Ditkin sets in hypergroups. Proc. Amer. Math. Soc. 77,353-358(USA) Mathematical Reviews 80g:43012 Zentralblatt fur Mathematik 388.43006(414.43005)
  3. Ajay Kumar and Ajit Iqbal Singh, (1981)Spectral synthesis in product and quotients of hypergroups. Pacific Journal of Mathematics 94,177-194(USA) Mathematical Reviews 82h:43005 Zentralblatt fur Mathematik 415.43010(458.43010)
  4. Wilfried Hauenschild, Eberhard Kaniuth and Ajay Kumar, (1983), Ideal structure of Beurling algebras on [FC]- groups. Journal of Functional Analysis 51,213-228(USA) Mathematical Reviews 84m:22007 Zentralblatt fur Mathematik 529.43005
  5. Wilfried Hauenschild, Eberhard Kaniuth and Ajay Kumar, (1984) Harmonic analysis on central hypergroups and induced representations. Pacific Journal of Mathematics 110,83-112(USA) Mathematical Reviews 85g:43015 Zentralblatt fur Mathematik 476.43007(528.43005)
  6. Ajay Kumar, (1986) Beurling algebras and structure of locally compact groups, Modern Analysis and its applications. Prentice Hall of India,229-240(India) Mathematical Reviews 89d:22008 Zentralblatt fur Mathematik 667.43003
  7. Ajay Kumar and Ajit Iqbal Singh, (1988) Counter examples in spectral synthesis on hypergroups. Rendiconti di Math. 8,147-155(Italy) Mathematical Reviews 91i:43004 Zentralblatt fur Mathematik 707.43003
  8. Ajay Kumar and Ajit Iqbal Singh, (1989) A dichotomy theorem for random walks on hypergroups. Lecture Notes in Mathematics, Springer Verlag 1379 179-184(Germany) Mathematical Reviews 90i:43004 Zentralblatt fur Mathematik 669.60017
  9. Oliver Gebuhrer and Ajay Kumar, (1989) Wiener property for a class of discrete hypergroups. Mathematische Zeitschrift 202,271-274 (Germany) Mathematical Reviews 90i:43004 Zentralblatt fur Mathematik 672.43004
  10. Siegfried Echterhoff, Eberhard Kaniuth and Ajay Kumar, (1991) Qualitative uncertainty principle for locally compact groups. Forum Mathematicum 3, 355-369(Germany) Mathematical Reviews 98a:43005 Zentralblatt fur Mathematik 725.43006
  11. Ajay Kumar, (1992) Qualitative uncertainty principle for hypergroups. Lecture Notes in Mathematics, Springer Verlag 1511,1-9(Germany) Mathematical Reviews 94a:43012 Zentralblatt fur Mathematik 781.43003
  12. Heinrich Begehr and Ajay Kumar, (1994) Bi-analytic functions of several complex variables. Complex Variables, Theory and Applications 24,89-106(USA) Mathematical Reviews 95k:32008 Zentralblatt fur Mathematik 794.32005
  13. Ajay Kumar, (1994) A generalized Riemann boundary problem in two variables. Arch. Math. 62,531-538 (Germany) Mathematical Reviews 95g:32995 Zentralblatt fur Mathematik 803.32001
  14. Ajay Kumar, (1994)Riemann Hilbert problem for a class of nth order systems. Complex Variables, Theory and Applications 25,11-22(USA) Mathematical Reviews 95k:30100 Zentralblatt fur Mathematik 804.30036
  15. Ajay Kumar and Allan M. Sinclair, (1998) Equivalence of norms on operator space tensor product of C*-algebras. Transactions of American Mathematical Society 350, 2033-2048 (USA) Mathematical Reviews 99a:46103 Zentralblatt fur Mathematik 906.46043
  16. Ajay Kumar, (2001) Qualitative uncertainty principle for certain hypergroups. Glasnik Mathematicki 36 ,33-38 (Crotia) Mathematical Reviews 2002h:43006 Zentralblatt fur Mathematik 982.43005
  17. Ajay Kumar, (2001) Operator space projective tensor product of C*-algebras. Mathematische Zeitschrift 237,211-217 (Germany) Mathematical Reviews 2002c:46114 Zentralblatt fur Mathematik 1035.46040
  18. Ajay Kumar, (2001) Involution and the Haagerup tensor product, Proc. Edinburgh Math. Soc. 44,317-322 (U.K.) Mathematical Reviews 2002c:46066 Zentralblatt fur Mathematik 1011.46051
  19. Eberhard Kaniuth and Ajay Kumar , (2001) Hardy’s Theorem for simply connected Nilpotent Lie groups. Math. Proc. Cambridge Phil. Soc. 131,487-494 (U.K.) Mathematical Reviews 2002c:22007 Zentralblatt fur Mathematik 996.43004
  20. Ajay Kumar and Chet Raj Bhatta, (2003) Uniform version of Wiener Tauberian theorem, Journal of Mathematical Sciences 2,63-71(India) Mathematical Reviews 2005h:43006 Zentralblatt fur Mathematik 1079.43006
  21. Ajay Kumar and Chet Raj Bhatta, (2004) An Uncertainty Principle like Hardy’s theorem for nilpotent Lie groups. Journal of Australian Math.Soc. 77,47-53 (Australia) Mathematical Reviews 2005f:43005 Zentralblatt fur Mathematik 1066.22006
  22. Ajay Kumar and Ravi Prakash, (2005) Boundary value problems for Poisson equation and bi-analytic functions, Complex Variables, Theory and Applications 50,597-608 (U.K.) Mathematical Reviews 2006c:30056 Zentralblatt fur Mathematik Pre 02247964
  23. Heinrich Begehr and Ajay Kumar, (2005) Boundary value problems for higher order inhomogeneous equations I, Analysis,International Mathematical journal of Analysis and its applications, 25,55-71(Germany) Mathematical Reviews 2006d:30067 Zentralblatt fur Mathematik 1077.30034
  24. Heinrich Begehr, Ajay Kumar, Dieter Schmersau and Judith C.Vanegas,(2005)Mixed complex boundary value problems in complex analysis, Intern. Connf. On Finite or Infinite Dimensional Complex Analysis and Applications. Eds. H. Kazama, M. Morimoto, C.C. Yang. Kyushu Univ. Press, Fukuoka, 25-40. Mathematical Reviews 2359680, Zentralblatt fur Mathematik 1129.30317.
  25. Ajay Kumar and Ravi Prakash, (2006) Mixed boundary value problems for higher order inhomogeneous equations, Complex Variables and Elliptic Equations 51,209-223 (USA), Mathematical Reviews 2006j:30086 Zentralblatt fur Mathematik 1092.30056
  26. Heinrich Begehr and Ajay Kumar, (2006) Boundary value problems for bi-polyanalytic functions, Applicable Analysis 85 1045-1077(USA), Mathematical Reviews 2007f:30073 Zentralblatt fur Mathematik 1105.30032 Cited- 8 times
  27. Heinrich Begehr and Ajay Kumar, (2007) Boundary value problems for higher order inhomogeneous equations II, Analysis, International Mathematical journal of Analysis and its applications, 27, 359-373 (Germany) Mathematical Reviews 2373661,Zentralblatt fur Mathematik 1137.30010
  28. Ajay Kumar and Ravi Prakash, (2007)Iterated Boundary Value Problems for the inhomogeneous Polyanalytic equation, Complex Variables and Elliptic Equations 52,921-932 (USA) Mathematical Reviews 2374962(2008m:30050) Zentralblatt fur Mathematik 1146.30031
  29. Ajay Kumar, (2007) Operator space structure of Banach Spaces, Math. Student 76,239-248 (India). Zentralblatt fur Mathematik pre 05575766 Mathematical Reviews 2522941
  30. Ajay Kumar and Ravi Prakash, (2008) Neumann and Mixed boundary value problem, Journal of Applied Functional Analysis 3, 399- 417 (USA). Mathematical Reviews 2387470(2009a:30096) Zentralblatt fur Mathematik 1165.30026
  31. Ajay Kumar and Ravi Prakash, (2008) Dirichlet problem for Inhomogeneous Polyharmonic equation, Complex Variables and Elliptic Equations 53, 643-651 (USA). Mathematical Reviews 2431345 ( 2009i : 31003) Zentralblatt fur Mathematik 1159.31302
  32. Ranjana Jain and Ajay Kumar, (2008) Operator space tensor product of C*-algebras, Mathematische Zeitschrift 260, 805-811 (Germany). Mathematical Reviews 2443331(2009i:46098) Zentralblatt fur Mathematik 1165.46030
  33. Ajay Kumar and Mukund Madhav Mishra, (2008) Polyharmonic Dirichlet problem on the Heisenberg group. Complex Variables and Elliptic Equations 53, 1103-1110. (USA). Mathematical Reviews 2467385 (2009m:35058) Zentralblatt fur Mathematik 1177.22005
  34. Arun Chaudhary and Ajay Kumar, (2009) Boundary value problems in the upper half plane. Complex Variables and Elliptic Equations 54(5), 441-448 (USA). Mathematical Reviews 2524139 ( 2010d:30054) Zentralblatt fur Mathematik 1166.30021
  35. Ajay Kumar and Ravi Prakash, (2009) Mixed boundary value problem for inhomogeneous Poly-analytic-harmonic equation, More Progress in Analysis- Proceedings of the 5th International ISAAC Congress, Catania Eds. H. Begehr, F. Niolosi ,World Scientific, Singapore,1149-1161. Zentralblatt fur Mathematik 1183.30041
  36. Heinrich Begehr, Arun Chaudhary and Ajay Kumar, (2010) Bi-polyanalytic functions in the upper half. Complex Variables and Elliptic Equations 55,305-316(USA). Mathematical Reviews 2011e;30111 Zentralblatt fur Mathematik 1188.30057
  37. Arun Chaudhary and Ajay Kumar, (2010) Mixed Boundary value problems in the upper half plane. Journal of Applied Functional Analysis 5, 209-220(USA). Zentralblatt fur Mathematik Pre 05775484 Mathematical Reviews 2675595 ( 2011g:30102)
  38. Ajay Kumar and Mukund Madhav Mishra, (2010) Green’s functions on the Heisenberg Group. Analysis, International Mathematical journal of Analysis and its applications 30,147-155(Germany) Mathematical Reviews 2604184(2011d;22008) Zentralblatt fur Mathematik 1197.22005
  39. Ranjana Jain and Ajay Kumar ,(2011) Ideals in operator space projective tensor product of C*-algebras, J. Aust. Math. Soc. 91, 275-288.
    Mathematical Reviews 2861849 Zbl pre05989540
  40. Ajay Kumar and Mukund Madhav Mishra (2012),Green functions and related boundary value problems on the Heisenberg group. To appear in Complex Variables and Elliptic Equations. Available online DOI:10.1080/17476933.2012.693482.
  41. Ajay Kumar and Mukund Madhav Mishra (2012), Powers of sub-laplacian on step two nilpotent Lie groups To appear in The Journal of Geometric Analysis. Available online DOI 10.1007/s12220-012-9298-0
  42. Ajay Kumar and Vandana Rajpal(2012) “Symmetry and quasi-centrality of operator space projective tensor product” Archiv der Mathematik. 99 , 519–529 (Germany) Zbl 1266.46044 Available also on http://arxiv.org/abs/1205.1679.
  43. Ajay Kumar and Mukund Madhav Mishra (2013),Green functions and related boundary value problems on the Heisenberg group. Complex Variables and Elliptic Equations 58, 547-556. Available online DOI:10.1080/17476933.2012.693482.
  44. Ajay Kumar and Mukund Madhav Mishra (2013), Powers of sub-laplacian on step two nilpotent Lie groups Journal of Geometric Analysis 23, 1559-1570. Available online DOI 10.1007/s12220-012-9298-0
  45. Ranjana Jain and Ajay Kumar, (2013) Spectral Synthesis for the Operator space projective tensor product of C*-algebras" Bulletin of the Malaysian Mathematical Sciences Society 36,855-864. Also available on . http://arxiv.org/abs/1108.3208.
  46. Ranjana Jain and Ajay kumar, (2014) Operator space tensor product of C*algebras: Embedding into second dual and ideal structure, Proc. Edinburgh Math. Soc.57.505-519. Available online DOI: http://dx.doi.org/10.1017/S001309151300045X 
  47. Ajay Kumar and Vandana Rajpal,(2014) -Regularity of Operator Space Projective Tensor Product of C*-Algebras” Journal of Applied Functional Analysis 9, 70-80.
  48. Ajay Kumar, (2014) From Fourier series to Harmonic analysis on locally compact groups, To appear in Math. Student
  49. Ajay Kumar and Vandana Rajpal, (2014)“ Projective tensor product of C*-algebras” To appear in Advances in Pure Mathematics Available on http://arxiv.org/abs/1305.0791.
  50. Vandana Rajpal, Ajay Kumar and Takashi Itoh,( 2014) "Schur tensor product of operator spaces” To appear in Forum Mathematicum Available on http://arxiv.org/abs/1308.4538.
  51. Shivani Dubey, Ajay Kumar and Mukund Madhav Mishra, Green's function for certain domains in the Heisenberg Group Hn,50. To appear in Boundary Value Problems.Available on http://arxiv.org/abs/1308.5643
  52. Ajay Kumar and Vandana Rajpal, Arens regularity of projective tensor products,51. Available on http://arxiv.org/abs/1305.0791v1
  53. Shivani Dubey, Ajay Kumar and Mukund Madhav Mishra,(2016) Neumann boundary value problem in domains of the Heisenberg Group Hn, To appear in Potential Analysis Published online http://dx.doi.org/10.1007/s11118 -016-9538-1. Also Available on http://arxiv.org/abs/1411.6838 ISSN:0926-2601(Print) 1572-929X(Online)
  54. Ashish Bansal, Ajay Kumar(2016) "Heisenberg Uncertainty Inequality for Gabor Transform" To appear in Journal of Mathematical Inequalities Available on http://arxiv.org/abs/1507.00446 ISSN: 1846-579X (print), 1848-9575 (online)
  55. Vandana Rajpal, Ajay Kumar, Lambda-tensor product of operator spaces, Available on http://arxiv.org/abs/1504.02846
  56. Ashish Bansal, Ajay Kumar, Qualitative Uncertainty Principle for Gabor transform” Available on http://arxiv.org/abs/1508.05475
  57. Shivani Dubey, Ajay Kumar and Mukund Madhav Mishra, Polyharmonic polynomials and mixed boundary value problems in the Heisenberg group Hn. Available on http://arxiv.org/abs/1511.00079.
  58. Preeti Luthra and Ajay Kumar, Embeddings and C*-envelopes of exact operator systems. Available on http://arxiv.org/abs/1603.01491
  59. Ashish Bansal, Ajay Kumar, Hardy's Theorem for Gabor transform Available on http://arxiv.org/abs/1603.02784

    Ph. D. Students
    • Chet Raj Bhatta, 2005, Behaviour of functions and their Fourier transforms.
    • Ravi Prakash, 2007, Boundary value problems in complex analysis.
    • Nisha Gupta, 2009, A study of Banach frames and related concepts in Banach spaces.
    • Arun Chaudhary, 2010, Complex Boundary value problems in unbounded regions.
    • Mukund Madhav Mishra, 2011, Potential theory of Stratified Lie groups.
    • Ranjana Jain, 2012, Operator Space Tensor Products of C*- algebras and their Ideal Structure.
    • Vandana, 2014, Projective norms on tensor products of operator spaces.
    • Sumit Nagpal, 2014, First and Second Order Differential Subordinations and Radius Problems for Caratheodary Functions
    • Jain, Sandhaya, 2015, Weighted function spaces of Lebesgue type.
    • Kumar, Ravinder, 2016, Efficient Numerical Algorithms for quasi-linear elliptic and hyperbolic partial differential equations.
    • Dubey, Shivani, 2016, Boundary value problems for the Kohn-Laplacian on the Heisenberg group Hn (submitted)
    • Bansal, Ashish, 2016, Heisenberg Inequality and Uncertainty Principle on locally compact groups, (submitted)
    • Preeti, Shivani, Monika Jain, Jyoti and Antony Janson (Current students).
    M. Phil Students
    • Verma ,Anita, 1999, The Grothendieck inequality.
    • Batra Rakesh,2005, Uncertainty principle on lovcally compact groups.
    • Vaid Ruchika, 2005, Higher order Cauchy-Pompeiu operators and their applications
    • Agarwal, Anu,2006, Arens regularity of certain Banach algebras.
    • Singhal Sandhya , 2006, Symmetry and Wiener property of weighted group algebras.
    • Khurana, Geetan, 2007, Ideal structure of regular Banach algebras.
    • Agarwal Shivani, 2008, Operator amenability of the Fourier algebra.
    • Kaur Gurpreet, 2008, Integral operators in Clifford analysis and polydomains.
    • Kukreja, Sapna,2009, Norm inequalities for operators on Hilbert spaces.
    • Kashyap Priyanka, 2009, Exact solutions of nonlinear differential equations.
    • Bansal Ashish,2010, Hardy's Uncertainty principle on nilpotent Lie groups.
    • Rajpal Vandana , 2011, Approximate identities and ideals in Banach algebras.
    • Rathi Poonam , 2012, Spectral Synthesis for Banach Algebras.
    • Preeti, 2013, Operator System structures and their tensor products.
    • Verma, Shelly, 2014, Bicomplex function theory and complex harmonic morphisms into bicomplex manifolds .
    • Pandey, Shesh Kumar, 2016, Boundary value problems on certain domains in the complex plane.

    Research Areas: The research work done can be divided in three major areas namely:
    1. Harmonic analysis on locally compact groups and hypergroups.
    2. Operator spaces tensor product of C*-algebras.
    3. Complex analytic methods in partial differential equations.

    1. The research work in harmonic analysis mainly deals with following problems:
      1. Ideal structure of group algebras.
      2. Spectral synthesis on convolution spaces of probability measures ( hypergroups).
      3. Uncertainty principle on Nilpotent Lie groups.
      4. Potential theory on Nilpotent Lie groups.
      In part (a) Wiener property, *-regularity of the Beurling algebras on [FC]- groups have been studied. In part (b), spectral synthesis of commutative hypergroup algebras including Wiener Tauberian theorem have been studied. Representation theory for hypergroups has been developed including Plancherel theorem, inversion theorem and Wiener property. In part (c), Hardy's theorem and qualitative uncertainty principle are proved for a variety of locally compact groups including connected nilpotent Lie groups. In part (d), representation formula for the higher powers of the sub- Laplacian by defining Green's function and Poisson kernel of higher order for H-type groups, which include the Heisenberg group have been given. The proofs exploit induced representations, Mackey theory, Kirillov's representation theory and harmonic analysis of nilpotent Lie groups and hypergroups. The work done has been published in Journal of Functional Analysis; Pacific J. Math.; Proc. Amer. Math. Soc.; Math. Zeit.; Forum Mathematicum; Math. Proc. Camb. Phil. Soc.; Complex Variables and Elliptic Equations; J.Aust.Math Soc.; Analysis; J. Geometric Analysis; Potential Analysis; J. Math. Inequalities; J. Inequalities and Applications.

    2. The theory of operator spaces grew out of the analysis of completely positive and completely bounded mappings. These maps were first studied on C*-algebras, and later on suitable subspaces of C*-algebras. The extension and representation theorems for completely bounded maps show that subspaces of C*-algebras carry an intrinsic metric structure which is preserved by completely isometries. Just as the theory of C*-algebras can be viewed as noncommutative topology and the theory of von Neumann algebras as noncommutative measure theory, one can think of the theory of operator spaces as noncommutative functional analysis. The operator space tensor product of C*-algebras particularly the Haagerup norm, operator space projective norm and Banach space projective norm have been investigated in the research work done. The ideal structure of *- Banach algebra of operator space tensor product of C*-algebras and of the Banach algebra of Haagerup tensor product have been studied. In particular, it has been shown that the Haagerup norm and Banach space projective norms are equivalent on the algebraic tensor product of two C*-algebras A and B if,and only if , either A or B is finite dimensional, or A and B are infinite dimensional and subhomogeneous. The same conclusion is proved if the Banach space projective norm is replaced by operator space projective norm. Involution on the Haagerup tensor product have been studied. It is shown that the centre of the operator space tensor product of two C*-algebras is the operator space tensor products of their centres. A characterization of isometric automorphisms of the two Banach algebras have been obtained. Methods involve the commutative and non-commutative Grothendieck inequalities, lifting maps to second duals and to tensor product of second duals and the structure of C*-algebras and von Neumann algebras. This work has been published in Trans.Amer.Math.Soc.; Math.Zeit.; Proc.Edin.Math.Soc.; Archiv der Mathematik; Forum Mathematicum; J. Australian Math. Soc.

    3. A systematic investigation of basic boundary value problems for complex partial differential of arbitrary order has been initiated. Integral representations are one of the important tools in analysis. Well-known representation the Cauchy's formula for analytic functions and Green's representation for harmonic functions. A variety of boundary value problems arising from Schwarz, Dirichlet, Neumann, Riemann Hilbert problems for the inhomogeneous polyanalytic equations have been investigated. Mixed boundary arising from a combination of Schwarz, Dirichlet and Neumann boundary value conditions have been studied for the inhomogeneous polyanalytic equations on the unit disc, half plane and quarter plane. Explicit representation formulas are obtained on these regions. Boundary value problems for bi-polyanalytic functions on different regions have been investigated. These investigations are based on Gauss theorem, Cauchy's Pompeiu representation, hierarchy of integral representations and iteration process leading from lower order equations to higher ones. The work has been published in Applicable analysis; Complex variables and Elliptic equations: Arch.Math.; Analysis; Journal of Applied Functional Analysis.