Scalar spectral measures associated with an operatorfractal
We study a spectraltheoretic model on a Hilbert space L{sup 2}(μ) where μ is a fixed Cantor measure. In addition to μ, we also consider an independent scaling operator U acting in L{sup 2}(μ). To make our model concrete, we focus on explicit formulas: We take μ to be the Bernoulli infiniteconvolution measure corresponding to scale number 1/4 . We then define the unitary operator U in L{sup 2}(μ) from a scaleby5 operation. The spectraltheoretic 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 projectionvalued measures and scalar spectral measures associated to U and its constituent parts. Our techniques make use of the representations of the Cuntz algebra O{sub 2} on L{sup 2}(μ)
 Authors:

^{[1]};
^{[2]};
^{[3]}
 Department of Mathematics, The University of Iowa, Iowa City, Iowa 522421419 (United States)
 Department of Mathematics, The University of Oklahoma, Norman, Oklahoma 730190315 (United States)
 Department of Mathematics and Statistics, Grinnell College, Grinnell, Iowa 501121690 (United States)
 Publication Date:
 OSTI Identifier:
 22251553
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 2; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; FRACTALS; HILBERT SPACE; SCALARS; SOCIOECONOMIC FACTORS