Functional calculus for generators of uniformly bounded holomorphic semigroups

1989 ◽  
Vol 38 (1) ◽  
pp. 91-103 ◽  
Author(s):  
Ralph de Laubenfels
Keyword(s):  
Functional Calculus ◽  
Holomorphic Semigroups ◽  
Bounded Holomorphic ◽  
Uniformly Bounded

scholarly journals A Besov algebra calculus for generators of operator semigroups and related norm-estimates

2019 ◽  
Author(s):  
Charles Batty ◽  
Alexander Gomilko ◽  
Yuri Tomilov
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Functional Calculus ◽  
The Other ◽  
Direct Approach ◽  
Operator Semigroups ◽  
Holomorphic Semigroups ◽  
Norm Estimates ◽  
Strict Extension ◽  
Bounded Holomorphic

Abstract We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille–Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.

scholarly journals Discrete quadratic estimates and holomorphic functional calculi in Banach spaces

1998 ◽  
Vol 58 (2) ◽  
pp. 271-290 ◽  
Author(s):  
Edwin Franks ◽  
Alan McIntosh
Keyword(s):  
Functional Calculus ◽  
Sufficient Conditions ◽  
Uniform Norm ◽  
Commuting Operators ◽  
Orthonormal Sequence ◽  
Quadratic Estimates ◽  
Holomorphic Functional Calculi ◽  
Necessary And Sufficient ◽  
Dyadic Decomposition ◽  
Uniformly Bounded

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.

scholarly journals Generation of generators of holomorphic semigroups

1993 ◽  
Vol 55 (2) ◽  
pp. 246-269 ◽  
Author(s):  
Christian Berg ◽  
Khristo Boyadzhiev ◽  
Ralph Delaubenfels
Keyword(s):  
Functional Calculus ◽  
Continuous Semigroup ◽  
Monotone Function ◽  
Linear Operators ◽  
Bernstein Function ◽  
Strongly Continuous Semigroup ◽  
Resolvent Set ◽  
The Relationship ◽  
Bounded Holomorphic ◽  
Unbounded Linear Operators

AbstractWe construct a functional calculus, g → g(A), for functions, g, that are the sum of a Stieltjes function and a nonnegative operator monotone function, and unbounded linear operators, A, whose resolvent set contains (−∞, 0), with {‖r(r + A)−1‖ ¦ r > 0} bounded. For such functions g, we show that –g(A) generates a bounded holomorphic strongly continuous semigroup of angle θ, whenever –A does.We show that, for any Bernstein function f, − f(A) generates a bounded holomorphic strongly continuous semigroup of angle π/2, whenever − A does.We also prove some new results about the Bochner-Phillips functional calculus. We discuss the relationship between fractional powers and our construction.

scholarly journals Functional Calculus for the Series of Semigroup Generators Via Transference

2019 ◽  
pp. 57-84
Author(s):  
Shawgy Hussein ◽  
Simon Joseph ◽  
Ahmed Sufyan ◽  
Murtada Amin ◽  
Ranya Tahire ◽  
...  
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Half Plane ◽  
Functional Calculus ◽  
Holomorphic Functions ◽  
Transference Principle ◽  
Semigroup Generators ◽  
Bounded Holomorphic ◽  
Semigroup Generator

In this paper, apply an established transference principle to obtain the boundedness of certain functional calculi for the sequence of semigroup generators. It is proved that if be the sequence generates 0- semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on right half–plane. The bounded of this calculus grows at most logarithmically as. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a so-called (strong) m-bounded calculus for all m∈ℕ, and that this property characterizes the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called semigroups, the Hilbert space results actually hold in general Banach spaces.

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