MathSciDoc: An Archive for Mathematician ∫

We prove that intermediate Banach spaces $$\mathcal{A}$$ and $$\mathcal{B}$$ with respect to arbitrary Hilbert couples $$\bar {H}$$ and $$\bar {K}$$ are exact interpolation if and onlyif they are exact$K$-monotonic, i.e. the condition $$f^0 \in \mathcal{A}$$ and the inequality $$K(t,g^0 ;\bar {K}) \leqslant K(t,f^0 ;\bar {H}),t > 0$$ , imply$g$^{0}∈$B$and ‖$g$^{0}‖$B$≤‖$f$^{0}‖_{$A$}($K$is Peetre’s$K$-functional). It is well known that this property is implied by the following: for each ϱ>1 there exists an operator $$T:\bar {H} \to \bar {K}$$ such that$Tf$^{0}=$g$^{0}, and $$K(t,Tf;\bar {K}) \leqslant \rho K(t,f;\bar {H}),f \in \mathcal{H}_0 + \mathcal{H}_1 ,t > 0$$ . Verifying the latter property, it suffices to consider the “diagonal case” where $$\bar {H} = \bar {K}$$ is finite-dimensional, in which case we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem it is shown that the statement remains valid when substituting ϱ=1. The result leads to a short proof of Donoghue’s theorem on interpolation functions, as well as Löwner’s theorem on monotone matrix functions.

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