Preprint
The spectral constant for the quantum cross and asymptotically sharp bounds for annuli
17 Apr 2025
Abstract
The quantum annulus of type $r$ is the class of invertible operators with
singular values in $(1/r,r).$ Given an analytic function on the classical
annulus of type $r,$ we may evaluate it on operators in the quantum annulus by
The spectral constant gives the maximum ratio betweeen the supremum over the
norm of evalutions at operators in the quantum annulus to the supremum over
classical evaluations. We show that the limit of the spectral constant as $r$
goes to infinity is $2.$ Via the correspondence between annuli and hyperbolae,
our study degenerates the problem to one on the quantum cross, pairs of
contractions with product zero, where the spectral constant is exactly $2.$
The essential technique is to rationally dilate $Z$ to $\hat{Z}$ which has $U
=(\hat{Z}+(\hat{Z}^{-1})^*)/(r+1/r)$ unitary and estimate $Uf(\hat{Z})U^*$
directly.
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Details
- Title
- The spectral constant for the quantum cross and asymptotically sharp bounds for annuli
- Creators
- J. E Pascoe
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022053486204721