ISSN:
1432-1785
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Calling a function f: R + p →R with $$\sum\limits_{i = 1}^n { \sum\limits_{j = 1}^n {\alpha _i \alpha _j } f(t_i + t_j )} \geqslant 0 for all (\alpha _1 ,...,\alpha _n ) \varepsilon R^n ,$$ , (t1,...,tn∈(R + p )n, n∈ℕ positiv semidefinit, the Laplace-transformations of finite nonnegative measures on R + p are charac terised as the continuous bounded positiv semidefinit functions. Let H be a real Hilbertspace. A σ-additive mapping $$M: \mathfrak{B}_ + ^p \to H$$ is called an orthogonal measure iff 〈M(A), M(B)⊃=0 for A∩B=ø. Exactly those mappings Y: R + p →H are Laplacetransformations of H-valued orthogonal measures, which are continuous and bounded and for which ⊂Y(s), Y(t)⊃ is only a function of s + t. Using this result one obtains a representation theorem for continuous semi-grouphomomorphisms defined on R + p with values in the “unit intervall” of the selfadjoint operators on H.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01411492
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