Title: Scalar spectral measures associated with an operator-fractal

We study a spectral-theoretic 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 infinite-convolution measure corresponding to scale number 1/4 . We then define the unitary operator U in L{sup 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{sub 2} on L{sup 2}(μ)

Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419 (United States)

Department of Mathematics, The University of Oklahoma, Norman, Oklahoma 73019-0315 (United States)

Department of Mathematics and Statistics, Grinnell College, Grinnell, Iowa 50112-1690 (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; SOCIO-ECONOMIC FACTORS