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Operator Valued Dirichlet Problem in the Plane

Year 2013, Volume: 42 Issue: 3, 269 - 280, 01.03.2013

Abstract

We consider an operator valued Dirichlet problem for harmonic mappings and prove the existence of a Perron-like solution. To formulatethe Perron’s construction we make use of Olson’s notion of spectralorder. We introduce a class of operator valued subharmonic mappingsand establish some of their elementary properties.

References

  • Akemann, C. A. and Weaver, N. Minimal upper bounds of commuting operators, Proc. Amer. Math. Soc. 124 (11) , 3469–3476, 1996.
  • Antezana, J. and Massey, P. and Stojanoff, D. Jensen’s inequality for spectral order and submajorization, J. Math. Anal. Appl. 331, 297–307, 2007.
  • Bonet, J. and Frerick, L. and Jord, E. Extension of vector-valued holomorphic and harmonic functions, Studia Math. 183 (3), 225–248, 2007.
  • Conway, J. B. A Course in operator theory, Grad. Texts in Math. 21, Amer. Math. Soc., 199 Enflo, P. and Smithies, L. Harnack’s theorem for harmonic compact operator-valued functions, Linear Algebra and its Applications 336, 21–27, 2001.
  • Fujii, M. and Kasahara, I. A remark on the spectral order of operators, Proc. Japan Acad. 47, 986–988, 1971.
  • Jord´ a, E. Vitali’s and Harnack’s type results for vector-valued functions, J. Math. Anal. Appl. 327, 739–743, 2007.
  • Olson, M. P. The selfadjoint operators of a Von Neumann algebra form a conditionally complete lattice, Proc. Amer. Soc. 28, 537–544, 1971.
  • Planeta, A. and Stochel, J. Spectral order for unbounded operators, J. Math. Anal. Appl., 1016/j.jmaa.2011.12.042.
  • Planeta, A. and Stochel, J. Multidimensional spectral order, preprint. Ransford, T. Potential theory in the complex plane, London Mathematical Society Student Texts 28, Cambridge University Press, Cambridge, 1995.

Operator Valued Dirichlet Problem in the Plane

Year 2013, Volume: 42 Issue: 3, 269 - 280, 01.03.2013

Abstract

-

References

  • Akemann, C. A. and Weaver, N. Minimal upper bounds of commuting operators, Proc. Amer. Math. Soc. 124 (11) , 3469–3476, 1996.
  • Antezana, J. and Massey, P. and Stojanoff, D. Jensen’s inequality for spectral order and submajorization, J. Math. Anal. Appl. 331, 297–307, 2007.
  • Bonet, J. and Frerick, L. and Jord, E. Extension of vector-valued holomorphic and harmonic functions, Studia Math. 183 (3), 225–248, 2007.
  • Conway, J. B. A Course in operator theory, Grad. Texts in Math. 21, Amer. Math. Soc., 199 Enflo, P. and Smithies, L. Harnack’s theorem for harmonic compact operator-valued functions, Linear Algebra and its Applications 336, 21–27, 2001.
  • Fujii, M. and Kasahara, I. A remark on the spectral order of operators, Proc. Japan Acad. 47, 986–988, 1971.
  • Jord´ a, E. Vitali’s and Harnack’s type results for vector-valued functions, J. Math. Anal. Appl. 327, 739–743, 2007.
  • Olson, M. P. The selfadjoint operators of a Von Neumann algebra form a conditionally complete lattice, Proc. Amer. Soc. 28, 537–544, 1971.
  • Planeta, A. and Stochel, J. Spectral order for unbounded operators, J. Math. Anal. Appl., 1016/j.jmaa.2011.12.042.
  • Planeta, A. and Stochel, J. Multidimensional spectral order, preprint. Ransford, T. Potential theory in the complex plane, London Mathematical Society Student Texts 28, Cambridge University Press, Cambridge, 1995.
There are 9 citations in total.

Details

Primary Language Turkish
Journal Section Mathematics
Authors

Nihat Gökhan Göğüş This is me

Publication Date March 1, 2013
Published in Issue Year 2013 Volume: 42 Issue: 3

Cite

APA Göğüş, N. G. (2013). Operator Valued Dirichlet Problem in the Plane. Hacettepe Journal of Mathematics and Statistics, 42(3), 269-280.
AMA Göğüş NG. Operator Valued Dirichlet Problem in the Plane. Hacettepe Journal of Mathematics and Statistics. March 2013;42(3):269-280.
Chicago Göğüş, Nihat Gökhan. “Operator Valued Dirichlet Problem in the Plane”. Hacettepe Journal of Mathematics and Statistics 42, no. 3 (March 2013): 269-80.
EndNote Göğüş NG (March 1, 2013) Operator Valued Dirichlet Problem in the Plane. Hacettepe Journal of Mathematics and Statistics 42 3 269–280.
IEEE N. G. Göğüş, “Operator Valued Dirichlet Problem in the Plane”, Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 3, pp. 269–280, 2013.
ISNAD Göğüş, Nihat Gökhan. “Operator Valued Dirichlet Problem in the Plane”. Hacettepe Journal of Mathematics and Statistics 42/3 (March 2013), 269-280.
JAMA Göğüş NG. Operator Valued Dirichlet Problem in the Plane. Hacettepe Journal of Mathematics and Statistics. 2013;42:269–280.
MLA Göğüş, Nihat Gökhan. “Operator Valued Dirichlet Problem in the Plane”. Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 3, 2013, pp. 269-80.
Vancouver Göğüş NG. Operator Valued Dirichlet Problem in the Plane. Hacettepe Journal of Mathematics and Statistics. 2013;42(3):269-80.