Operator Valued Dirichlet Problem in the Plane

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

Nihat Gökhan Göğüş

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.

Keywords

Operator theory, Harmonic mappings, Perron method, Spectral order.

Operator Valued Dirichlet Problem in the Plane

Abstract

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Keywords

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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.