Sectorial operator

Last updated September 02, 2024

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.

Definition

Let $(X,\|\cdot \|)$ be a Banach space. Let $A$ be a (not necessarily bounded) linear operator on $X$ and $\sigma (A)$ its spectrum.

For the angle $0<\omega \leq \pi$ , we define the open sector

\Sigma _{\omega }:=\{z\in \mathbb {C} \setminus \{0\}:|\operatorname {arg} z|<\omega \}

,

and set $\Sigma _{0}:=(0,\infty )$ if $\omega =0$ .

Now, fix an angle $\omega \in [0,\pi )$ . The operator $A$ is called sectorial with angle $\omega$ if^[1]

\sigma (A)\subset {\overline {\Sigma _{\omega }}}

and if

\sup \limits _{\lambda \in \mathbb {C} \setminus {\overline {\Sigma _{\psi }}}}|\lambda |\|(\lambda -A)^{-1}\|<\infty

for every larger angle $\psi \in (\omega ,\pi )$ . The set of sectorial operators with angle $\omega$ is denoted by $\operatorname {Sect} (\omega )$ .

Remarks

If $\omega \neq 0$ , then $\Sigma _{\omega }$ is open and symmetric over the positive real axis with angular aperture $2\omega$ .

Bibliography

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

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References

↑ 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.

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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.

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