Vol. 21, No. 1, 1967

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On the convergence of resolvents of operators

Minoru Hasegawa

Vol. 21 (1967), No. 1, 35–47

Let a family of linear operators {An}(n = 1,2,) in a Banach space X have the resolvents {R(λ;An)} which is equicontinuous in n. Suppose that {An} is a Cauchy sequence on a dense set. Then the question of convergence arises; when will {R(λ;An)x} be a Cauchy sequence for all x X?

This problem is treated in some special cases and an application to the following theorem is presented.

Let A be the generator of a positive contraction semigroup and let B be a linear operator with domain 𝒟(B) ⊃𝒟(A) in a weakly complete Banach lattice X.

Then A + B or its closed extension generates a positive contraction semi-group Σwhich dominates Σ if and only if A + B is dissipative and B is positive.

Mathematical Subject Classification
Primary: 47.50
Received: 27 October 1965
Published: 1 April 1967
Minoru Hasegawa