Study of the boundary of numerrical range of Hilbert or Krein space operators and its applications

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Some interesting results are found on the boundary of numerical range of linear operators in a finite dimensional Hilbert space. Especially some algebraic curve theoretic analysis is performed for shift operators. Concerning with Krein space numerical ranges, some results on the product ranges are obtained. The result will contribute to the basis of applications in nuclear magnetic resonance or spectral scopy. The results are presented in 16 academic papers and one book. By 7 presentations, the results are presented. New results in dynamics and hyperbolic differential equations are also obtained.