Interpolation in self-adjoint settings

Authors:
Y. S. Jo, J. H. Kang, R. L. Moore and T. T. Trent

Journal:
Proc. Amer. Math. Soc. **130** (2002), 3269-3281

MSC (2000):
Primary 46L10, 47L35

DOI:
https://doi.org/10.1090/S0002-9939-02-06610-8

Published electronically:
June 11, 2002

MathSciNet review:
1913006

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the operator equation , where the operators and are given and the operator is required to lie in some von Neumann algebra. We derive a necessary and sufficient condition for the existence of a solution . The condition is that there must exist a constant so that, for all finite collections of operators in the commutant, and all collections of vectors , we have We also study the *equality* , in connection with solving the equation where the operator is required to lie in some CSL algebra.

**1.**W. Arveson,*Operator algebras and invariant subspaces*, Annals of Math.**100**(3) (1974), 433-532. MR**51:1420****2.**R. G. Douglas,*On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space*, Proceedings of American Mathematical Society**17**(1966), 413-415. MR**34:3315****3.**A. Hopenwasser,*The Equation in a Reflexive Operator Algebra*, Indiana University Mathematics Journal**29**(1980), 121-126. MR**81c:47014****4.**E. Katsoulis,*Remarks on the Interpolation and the Similarity Problem for Nest Subalgebras of von Neumann Algebras*, Journal of Mathematical Analysis and Applications**190**(1995), 755-762. MR**96b:47051****5.**E.C. Lance,*Some Properties of Nest Algebras*, Proceedings of the London Mathematical Society (3)**19**(1969), 45-68. MR**39:3325****6.**H. Meschkowski,*Hilbertsche Räume mit Kernfunktion*, Springer Verlag, Berlin, 1962. MR**25:4326****7.**R.L. Moore and T.T. Trent,*Solving Operator Equations in Nest Algebras*, Houston Journal of Mathematics**24**(1998), 483-488. MR**2000a:47149****8.**R.L. Moore and T.T. Trent,*Factoring Positive Operators on Reproducing Kernel Hilbert Spaces*, Journal of Integral Equations and Operator Theory**24**(1996), 470-483. MR**97f:47017**

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Additional Information

**Y. S. Jo**

Affiliation:
Department of Mathematics, Keimyung University, Taegu, Korea

**J. H. Kang**

Affiliation:
Department of Mathematics, Taegu University, Taegu 712-714, Korea

**R. L. Moore**

Affiliation:
Department of Mathematics, Box 870350, University of Alabama, Tuscaloosa, Alabama 35487-0350

Email:
rmoore@gp.as.ua.edu

**T. T. Trent**

Affiliation:
Department of Mathematics, Box 870350, University of Alabama, Tuscaloosa, Alabama 35487-0350

Email:
ttrent@gp.as.ua.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06610-8

Received by editor(s):
September 1, 2000

Received by editor(s) in revised form:
June 7, 2001

Published electronically:
June 11, 2002

Communicated by:
David R. Larson

Article copyright:
© Copyright 2002
American Mathematical Society