holomorphic functional calculi
Recently Published Documents


TOTAL DOCUMENTS

7
(FIVE YEARS 0)

H-INDEX

4
(FIVE YEARS 0)

2005 ◽  
Vol 48 (2) ◽  
pp. 423-444 ◽  
Author(s):  
Markus Haase

AbstractWe present an abstract approach to the construction of holomorphic functional calculi for unbounded operators and apply it to the special case of sectorial operators. In effect, we obtain a calculus for a much larger class of functions than was known before, including certain meromorphic functions. We discuss the role of topology. Then we prove in detail a composition rule $(f\circ g)(A)=f(g(A))$ which is the main result of the paper. This is done in such a way that the proof can easily be transferred to functional calculi for other classes of operators.


2003 ◽  
Vol 67 (2) ◽  
pp. 187-200 ◽  
Author(s):  
Xuan Thinh Duong ◽  
Lixin Yan

Let χ be a space of homogeneous type of infinite measure. Let T be a singular integral operator which is bounded on Lp (χ) for some p, 1 < p < ∞. We give a sufficient condition on the kernel of T so that when a function b ∈ BMO(χ), the commutator [b, T](f) = T (bf) – bT (f) is bounded on Lp spaces for all p, 1 < p > ∞. Our condition is weaker than the usual Hörmander condition. Applications include Lp-boundedness of the commutators of BMO functions and holomorphic functional calculi of Schrödinger operators, and divergence form operators on irregular domains.


1998 ◽  
Vol 58 (2) ◽  
pp. 271-290 ◽  
Author(s):  
Edwin Franks ◽  
Alan McIntosh

We develop a discrete version of the weak quadratic estimates for operators of type w explained by Cowling, Doust, McIntosh and Yagi, and show that analogous theorems hold. The method is direct and can be generalised to the case of finding necessary and sufficient conditions for an operator T to have a bounded functional calculus on a domain which touches σ(T) nontangentially at several points. For operators on Lp, 1 < p < ∞, it follows that T has a bounded functional calculus if and only if T satisfies discrete quadratic estimates. Using this, one easily obtains Albrecht's extension to a joint functional calculus for several commuting operators. In Hilbert space the methods show that an operator with a bounded functional calculus has a uniformly bounded matricial functional calculus.The basic idea is to take a dyadic decomposition of the boundary of a sector Sv. Then on the kth ingerval consider an orthonormal sequence of polynomials . For h ∈ H∞(Sν), estimates for the uniform norm of h on a smaller sector Sμ are obtained from the coefficients akj = (h, ek, j). These estimates are then used to prove the theorems.


1998 ◽  
Vol 58 (2) ◽  
pp. 291-305 ◽  
Author(s):  
David Albrecht ◽  
Edwin Franks ◽  
Alan McIntosh

Let S and T be commuting operators of type ω and type ϖ in a Banach space X. Then the pair has a joint holomorphic functional calculus in the sense that it is possible to define operators f(S, T) in a consistent manner, when f is a suitable holomorphic function defined on a product of sectors. In particular, this gives a way to define the sum S + T when ω + ϖ < π. We show that this operator is always of type μ where μ = max{ω, ϖ}. We explore when bounds on the individual functional calculi of S and T imply bounds on the functional calculus of the pair (S, T), and some implications for the regularity problem of when ∥(S + T)u∥ is equivalent to ∥Su∥ + ∥Tu∥.


Sign in / Sign up

Export Citation Format

Share Document