Abstract
If exp {−tH}, exp {−tK}, are self-adjoint, positivity preserving, contraction semigroups on a Hilbert space ℋ=L 2(X;dμ) we write
whenever exp {−tH}-exp {−tK} is positivity preserving for allt≧0 and then we characterize the class of positive functions for which (*) always implies
This class consists of thef∈C ∞(0, ∞) with
In particular it contains the class of monotone operator functions. Furthermore if exp {−tH} isL p(X;dμ) contractive for allp∈[1, ∞] and allt>0 (or, equivalently, forp=∞ andt>0) then exp {−tf(H)} has the same property. Various applications to monotonicity properties of Green's functions are given.
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Communicated by H. Araki
On leave of absence from the University of Oslo
On leave of absence from Yokohama City University
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Bratteli, O., Kishimoto, A. & Robinson, D.W. Positivity and monotonicity properties ofC 0-semigroups. I. Commun.Math. Phys. 75, 67–84 (1980). https://doi.org/10.1007/BF01962592
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DOI: https://doi.org/10.1007/BF01962592