TY - JOUR

T1 - Laplace and laplace-stieltjes space

AU - DeLaubenfels, Ralph D.

AU - Kantorovitz, Shmuel

PY - 1993

Y1 - 1993

N2 - Suppose X is a Banach space and f(hook) is a function from [0,∞) into the space of all (possibly unbounded) linear operators on X. We construct a maximal Banach subspace, Y, the Laplace space, continuously embedded in X, so that f(hook)|Y is the Laplace transform of a strongly continuous family of contractions on Y, and a maximal Banach subspace, W, the Laplace-Stieltjes space, continuously embedded in X, so that the map s (mapping)f(hook)(s)x is the Laplace-Stieltjes transform of a vector-valued measure ∀x∈W. Under appropriate conditions of f(hook), that are satisfied in the eases of most interest, the Laplace space contains all x so that the map s(mapping)f(hook)(s)x is a uniformly strongly continuous Laplace transform. Appropriate choices of f(hook) yield maximal subspaces on which integrodifferential equations or abstract Cauchy problems, of arbitrarily high order, are well-posed. Other choices of f(hook) produce the semisimplicity manifold and the maximal subspace on which an operator is well-bounded. In the former case, the space contains all initial data for which the abstract Cauchy problem has a solution that equals the Laplace-Stieltjes transform of a vector-valued measure.

AB - Suppose X is a Banach space and f(hook) is a function from [0,∞) into the space of all (possibly unbounded) linear operators on X. We construct a maximal Banach subspace, Y, the Laplace space, continuously embedded in X, so that f(hook)|Y is the Laplace transform of a strongly continuous family of contractions on Y, and a maximal Banach subspace, W, the Laplace-Stieltjes space, continuously embedded in X, so that the map s (mapping)f(hook)(s)x is the Laplace-Stieltjes transform of a vector-valued measure ∀x∈W. Under appropriate conditions of f(hook), that are satisfied in the eases of most interest, the Laplace space contains all x so that the map s(mapping)f(hook)(s)x is a uniformly strongly continuous Laplace transform. Appropriate choices of f(hook) yield maximal subspaces on which integrodifferential equations or abstract Cauchy problems, of arbitrarily high order, are well-posed. Other choices of f(hook) produce the semisimplicity manifold and the maximal subspace on which an operator is well-bounded. In the former case, the space contains all initial data for which the abstract Cauchy problem has a solution that equals the Laplace-Stieltjes transform of a vector-valued measure.

UR - http://www.scopus.com/inward/record.url?scp=0002199412&partnerID=8YFLogxK

U2 - 10.1006/jfan.1993.1103

DO - 10.1006/jfan.1993.1103

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SN - 0022-1236

VL - 116

SP - 1

EP - 61

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 1

ER -