50 Years with Hardy Spaces pp 83-95 | Cite as

# Interpolation by the Derivatives of Operator Lipschitz Functions

## Abstract

Let Λ be a discrete subset of the real line ℝ. We prove that for every bounded function *φ* on Λ there exists an operator Lipschitz function *f* on ℝ such that *f’* (*t*) = *φ*(*t*) for all *t∈*Λ. The same is true for the set of operator Lipschitz functions *f* on ℝ such that *f’* coincides with the non-tangential boundary values of a bounded holomorphic function on the upper half-plane. In other words, for every bounded function *φ* on Λ there exists a commutator Lipschitz function *f* on the closed upper half-plane such that *f’* (*t*) = *φ*(*t*) for all *t∈*Λ. The same is also true for some non-discrete countable sets Λ. Furthermore, we consider the case where Λ is a subset of the closed upper half-plane, Λ*⊄* ℝ. Similar questions for commutator Lipschitz functions on a closed subset F of ℂ are also considered.

## Keywords

Operator Lipschitz functions commutator Lipschitz functions interpolation## Mathematics Subject Classification (2010)

Primary 47A56 Secondary 30H05## Preview

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