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Sectorial operator

From Wikipedia, the free encyclopedia

In mathematics, more precisely in operator theory, a sectorial operator is a linear operator on a Banach space whose spectrum in an open sector in the complex plane and whose resolvent is uniformly bounded from above outside any larger sector. Such operators might be unbounded.

Sectorial operators have applications in the theory of elliptic and parabolic partial differential equations.

Definition

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Let be a Banach space. Let be a (not necessarily bounded) linear operator on and its spectrum.

For the angle , we define the open sector

,

and set if .

Now, fix an angle . The operator is called sectorial with angle if[1]

and if

for every larger angle . The set of sectorial operators with angle is denoted by .

Remarks

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  • If , then is open and symmetric over the positive real axis with angular aperture .

Bibliography

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  • Markus Haase (2006), Birkhäuser Basel (ed.), The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169, doi:10.1007/3-7643-7698-8, ISBN 978-3-7643-7697-0
  • Atsushi Yagi (2010), "Sectorial Operators", Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics, Berlin, Heidelberg: Springer, pp. 55–116, doi:10.1007/978-3-642-04631-5_2, ISBN 978-3-642-04630-8
  • Markus Haase (2003), Universität Ulm (ed.), The Functional Calculus for Sectorial Operators and Similarity Methods

References

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  1. ^ Haase, Markus (2006). The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications. p. 19. doi:10.1007/3-7643-7698-8. ISBN 978-3-7643-7697-0.