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A Note on Sectorial and R-Sectorial Operators

Alberto Venni — 2008

Bollettino dell'Unione Matematica Italiana

The following results are proved: (i) if α , β ∈ ℝ + and A is a sectorial operator, then the set { t α ⁢ A β ⁢ ( t + A ) ; t > 0 } is bounded; (ii) the same set of operators is R-bounded if A is R-sectorial.

H ∞ functional calculus for sectorial and bisectorial operators

Giovanni Dore;  Alberto Venni — 2005

Studia Mathematica

We give a concise exposition of the basic theory of H ∞ functional calculus for N-tuples of sectorial or bisectorial operators, with respect to operator-valued functions; moreover we restate and prove in our setting a result of N. Kalton and L. Weis about the boundedness of the operator f ( T ₁ , . . . , T N ) when f is an R-bounded operator-valued holomorphic function.

H ∞ functional calculus for an elliptic operator on a half-space with general boundary conditions

Giovanni Dore;  Alberto Venni — 2002

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let A be the L p realization ( 1 < p < ∞ ) of a differential operator P ( D x , D t ) on ℝ n × ℝ + with general boundary conditions B k ( D x , D t ) u ( x , 0 ) = 0 ( 1 ≤ k ≤ m ). Here P is a homogeneous polynomial of order 2 m in n + 1 complex variables that satisfies a suitable ellipticity condition, and for 1 ≤ k ≤ m B k is a homogeneous polynomial of order m k < 2 m ; it is assumed that the usual complementing condition is satisfied. We prove that A is a sectorial operator with a bounded H ∞ functional calculus.

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