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The behavior of the essential spectrum and the essential norm under (complex/real) interpolation is investigated. We extend an example of Albrecht and Müller for the spectrum by showing that in complex interpolation the essential spectrum of an interpolated operator is also in general a discontinuous map of the parameter θ. We discuss the logarithmic convexity (up to a multiplicative constant) of the essential norm under real interpolation, and show that this holds provided certain compact approximation conditions are satisfied. Some evidence supporting a counterexample is presented.
A. G. Aksoy, and H.-O. Tylli. "Interpolation of the essential spectrum and the essential norm." Banach Center Publications 68.1 (2005): 9-18. <http://eudml.org/doc/282005>.
@article{A2005, abstract = {The behavior of the essential spectrum and the essential norm under (complex/real) interpolation is investigated. We extend an example of Albrecht and Müller for the spectrum by showing that in complex interpolation the essential spectrum $σ_e(S_\{[θ]\})$ of an interpolated operator is also in general a discontinuous map of the parameter θ. We discuss the logarithmic convexity (up to a multiplicative constant) of the essential norm under real interpolation, and show that this holds provided certain compact approximation conditions are satisfied. Some evidence supporting a counterexample is presented.}, author = {A. G. Aksoy, H.-O. Tylli}, journal = {Banach Center Publications}, keywords = {essential spectrum; essential norm; complex interpolation; real interpolation}, language = {eng}, number = {1}, pages = {9-18}, title = {Interpolation of the essential spectrum and the essential norm}, url = {http://eudml.org/doc/282005}, volume = {68}, year = {2005}, }
TY - JOUR AU - A. G. Aksoy AU - H.-O. Tylli TI - Interpolation of the essential spectrum and the essential norm JO - Banach Center Publications PY - 2005 VL - 68 IS - 1 SP - 9 EP - 18 AB - The behavior of the essential spectrum and the essential norm under (complex/real) interpolation is investigated. We extend an example of Albrecht and Müller for the spectrum by showing that in complex interpolation the essential spectrum $σ_e(S_{[θ]})$ of an interpolated operator is also in general a discontinuous map of the parameter θ. We discuss the logarithmic convexity (up to a multiplicative constant) of the essential norm under real interpolation, and show that this holds provided certain compact approximation conditions are satisfied. Some evidence supporting a counterexample is presented. LA - eng KW - essential spectrum; essential norm; complex interpolation; real interpolation UR - http://eudml.org/doc/282005 ER -