Local Dirichlet spaces

as de Branges-Rovnyak spaces

Author:
Donald Sarason

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2133-2139

MSC (1991):
Primary 46E20

DOI:
https://doi.org/10.1090/S0002-9939-97-03896-3

MathSciNet review:
1396993

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Abstract | References | Similar Articles | Additional Information

Abstract: The harmonically weighted Dirichlet spaces associated with unit point masses are shown to coincide with de Branges-Rovnyak spaces, with equality of norms. As a consequence it is shown that radial expansion operators act contractively in harmonically weighted Dirichlet spaces.

**1.**L. de Branges and J. Rovnyak,*Square Summable Power Series*, Holt, Rinehart, and Winston, New York, 1966. MR**35:5909****2.**B. A. Lotto and D. Sarason,*Multipliers of de Branges-Rovnyak spaces*, Indiana Univ. Math. J.**42**(1993), 907-920. MR**95a:46039****3.**S. Richter,*A representation theorem for cyclic analytic two-isometries*, Trans. Amer. Math. Soc.**328**(1991), 325-349. MR**92e:47052****4.**S. Richter and C. Sundberg,*A formula for the local Dirichlet integral*, Michigan Math. J.**38**(1991), 355-379. MR**92i:47035****5.**S. Richter and C. Sundberg,*Multipliers and invariant subspaces in the Dirichlet space*, J. Operator Theory**28**(1992), 167-186. MR**95e:47007****6.**S. Richter and C. Sundberg,*Invariant subspaces of the Dirichlet shift and pseudocontinuations*, Trans. Amer. Math. Soc.**341**(1994), 863-879. MR**94d:47026****7.**D. Sarason,*Sub-Hardy Hilbert Spaces in the Unit Disk*, Wiley, New York, 1995.**8.**D. Sarason,*Harmonically weighted Dirichlet spaces associated with finitely atomic measures*, in preparation.

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Additional Information

**Donald Sarason**

Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720

Email:
sarason@math.berkeley.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03896-3

Received by editor(s):
February 13, 1996

Communicated by:
Theodore W. Gamelin

Article copyright:
© Copyright 1997
American Mathematical Society