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Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform

Ralph deLaubenfels — 1992

Studia Mathematica

Suppose A is a (possibly unbounded) linear operator on a Banach space. We show that the following are equivalent. (1) A is well-bounded on [0,∞). (2) -A generates a strongly continuous semigroup e - s A s 0 such that ( 1 / s 2 ) e - s A s > 0 is the Laplace transform of a Lipschitz continuous family of operators that vanishes at 0. (3) -A generates a strongly continuous differentiable semigroup e - s A s 0 and ∃ M < ∞ such that H n ( s ) ( k = 0 n ( s k A k ) / k ! ) e - s A M , ∀s > 0, n ∈ ℕ ∪ 0. (4) -A generates a strongly continuous holomorphic semigroup e - z A R e ( z ) > 0 that is O(|z|) in all...

Automatic extensions of functional calculi

Ralph deLaubenfels — 1995

Studia Mathematica

Given a Banach algebra ℱ of complex-valued functions and a closed, linear (possibly unbounded) densely defined operator A, on a Banach space, with an ℱ functional calculus we present two ways of extending this functional calculus to a much larger class of functions with little or no growth conditions. We apply this to spectral operators of scalar type, generators of bounded strongly continuous groups and operators whose resolvent set contains a half-line. For f in this larger class, one construction...

Inverses of generators of nonanalytic semigroups

Ralph deLaubenfels — 2009

Studia Mathematica

Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup e t A t 0 . It is shown that A - 1 generates an O ( 1 + τ ) A ( 1 - A ) - 1 -regularized semigroup. Several equivalences for A - 1 generating a strongly continuous semigroup are given. These are used to generate sufficient conditions on the growth of e t A t 0 , on subspaces, for A - 1 generating a strongly continuous semigroup, and to show that the inverse of -d/dx on the closure of its image in L¹([0,∞)) does not generate a strongly...

Decomposable embeddings, complete trajectories, and invariant subspaces

Ralph deLaubenfelsPhóng Vũ — 1996

Studia Mathematica

We produce closed nontrivial invariant subspaces for closed (possibly unbounded) linear operators, A, on a Banach space, that may be embedded between decomposable operators on spaces with weaker and stronger topologies. We show that this can be done under many conditions on orbits, including when both A and A* have nontrivial non-quasi-analytic complete trajectories, and when both A and A* generate bounded semigroups that are not stable.

Functional calculi, regularized semigroups and integrated semigroups

Ralph deLaubenfelsMustapha Jazar — 1999

Studia Mathematica

We characterize closed linear operators A, on a Banach space, for which the corresponding abstract Cauchy problem has a unique polynomially bounded solution for all initial data in the domain of A n , for some nonnegative integer n, in terms of functional calculi, regularized semigroups, integrated semigroups and the growth of the resolvent in the right half-plane. We construct a semigroup analogue of a spectral distribution for such operators, and an extended functional calculus: When the abstract...

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