Abstract
We study joint numerical and spectral radii defined for d-tuples of bounded operators on a Hilbert space and related to noncommutative notions of independence. The definitions are in analogy with the ones of Popescu, where his formulations turned out to be related with free creation operators, and in this way related to the free independence of Voiculescu. In our study the definitions are related with either weakly monotone creation operators, and thus associated with the monotone independence of Muraki, or with boolean creation operators, and hence related with the boolean independence.
Highlights
The notion of numerical radius as well as the related notion of numerical range is an object of intensive studies since the work by Toeplitz [10] in 1918 until today
The main idea of this paper is to study analogues of these definitions in the case where we replace the full Fock space by a Fock space associated to other noncommutative independences, and the free creation operators by the creation operators on the appropriate Fock space
This paper is a beginning of the studies of joint numerical and spectral radii related to creation operators independent in noncommutative sense: the monotone and boolean ones
Summary
The notion of numerical radius as well as the related notion of numerical range is an object of intensive studies since the work by Toeplitz [10] in 1918 until today. The (free) joint spectral radius is rFðT1; . These two notions are related to the free semigroup Fþd , and to the model of freeness of Voiculescu [11], and more precisely to the creation operators on the full Fock space by the following result. Theorem 3 ([9], Corollary 1.2) The joint free numerical and spectral radii can be computed as the ordinary numerical radius w and the spectral radius r of single operators: wFðT1; . We show that the joint (noncommutative) numerical radii, defined in analogy to Theorem 1, satisfy many basic properties similar to the free case. We show the unitarity invariance and the relation with the appropriately defined spectral radius. In this paper there is no need to define the monotone and boolean independences, it is sufficient to consider the models of both of them, built on either weakly monotone Fock space or on the Boolean Fock space, respectively
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