Journal article
Scalar spectral measures associated with an operator-fractal
Journal of mathematical physics, Vol.55(2), p.22103
2014
DOI: 10.1063/1.4863897
Abstract
We study a spectral-theoretic model on a Hilbert space L 2(μ) where μ is a fixed Cantor measure. In addition to μ, we also consider an independent scaling operator U acting in L 2(μ). To make our model concrete, we focus on explicit formulas: We take μ to be the Bernoulli infinite-convolution measure corresponding to scale number 1 4 . We then define the unitary operator U in L 2(μ) from a scale-by-5 operation. The spectral-theoretic and geometric properties we have previously established for U are as follows: (i) U acts as an ergodic operator; (ii) the action of U is not spatial; and finally, (iii) U is fractal in the sense that it is unitarily equivalent to a countable infinite direct sum of (twisted) copies of itself. In this paper, we prove new results about the projection-valued measures and scalar spectral measures associated to U and its constituent parts. Our techniques make use of the representations of the Cuntz algebra O 2 on L 2(μ).
Details
- Title: Subtitle
- Scalar spectral measures associated with an operator-fractal
- Creators
- Palle E.T JorgensenKeri A KornelsonKaren L Shuman
- Resource Type
- Journal article
- Publication Details
- Journal of mathematical physics, Vol.55(2), p.22103
- DOI
- 10.1063/1.4863897
- ISSN
- 0022-2488
- eISSN
- 1089-7658
- Grant note
- DOI: 10.13039/100000001, name: NSF, award: DMS-0701164
- Language
- English
- Date published
- 2014
- Academic Unit
- Mathematics
- Record Identifier
- 9983986091402771
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