Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

A decomposition of weighted translation operators
HTML articles powered by AMS MathViewer

by Joseph J. Bastian PDF
Trans. Amer. Math. Soc. 224 (1976), 217-230 Request permission

Abstract:

Let (X, A, m) be a measure space and T an invertible measure-preserving transformation on X. Given $\phi$ in ${L^\infty }(X)$, define operators ${M_\phi }$ and ${U_T}$ on ${L^2}(X)$ by $({M_\phi }f)(x) = \phi (x)f(x)$ and $({U_T}f) = f(Tx)$. Operators of the form ${M_\phi }{U_T}$ are called weighted translation operators. In this paper it is shown that every weighted translation operator on a sufficiently regular measure space an be decomposed into a direct integral of weighted translation operators where almost all of the transformations in the integrand are ergodic. It is also shown that every hyponormal weighted translation operator defined by an ergodic transformation is either normal or unitarily equivalent to a bilateral weighted shift. These two results along with some results concerning direct integrals of hyponormal and subnormal operators are used to show that every hyponormal (resp. subnormal) weighted translation operator is unitarily equivalent to a direct integral of normal operators and hyponormal (resp. subnormal) bilateral weighted shifts. The paper concludes with an example.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B37, 47A35
  • Retrieve articles in all journals with MSC: 47B37, 47A35
Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 224 (1976), 217-230
  • MSC: Primary 47B37; Secondary 47A35
  • DOI: https://doi.org/10.1090/S0002-9947-1976-0420327-8
  • MathSciNet review: 0420327