Operator realizations of non-commutative analytic functions
A realization is a triple, $(A,b,c)$ , consisting of a $d-$ tuple, $A= (A_1, \cdots , A_d )$ , $d\in \mathbb {N}$ , of bounded linear operators on a separable, complex Hilbert space, $\mathcal {H}$ , and vectors $b,c \in \mathcal {H}$ . Any such realization define...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100388/type/journal_article |
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| Summary: | A realization is a triple,
$(A,b,c)$
, consisting of a
$d-$
tuple,
$A= (A_1, \cdots , A_d )$
,
$d\in \mathbb {N}$
, of bounded linear operators on a separable, complex Hilbert space,
$\mathcal {H}$
, and vectors
$b,c \in \mathcal {H}$
. Any such realization defines an analytic non-commutative (NC) function in an open neighbourhood of the origin,
$0:= (0, \cdots , 0)$
, of the NC universe of
$d-$
tuples of square matrices of any fixed size. For example, a univariate realization, i.e., where A is a single bounded linear operator, defines a holomorphic function of a single complex variable, z, in an open neighbourhood of the origin via the realization formula
$b^{*} (I-zA)^{-1} c$
. |
|---|---|
| ISSN: | 2050-5094 |