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Factorisation in nest algebras. II

Authors: M. Anoussis and E. G. Katsoulis
Journal: Trans. Amer. Math. Soc. 350 (1998), 165-183
MSC (1991): Primary 47D25
MathSciNet review: 1451593
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Abstract: The main result of this paper is Theorem 5, which provides a necessary and sufficient condition on a positive operator $A$ for the existence of an operator $B$ in the nest algebra $AlgN$ of a nest $N$ satisfying $A=BB^{*}$ (resp. $A=B^{*}B)$. In Section 3 we give a new proof of a result of Power concerning outer factorisation of operators. We also show that a positive operator $A$ has the property that there exists for every nest $N$ an operator $B_N$ in $AlgN$ satisfying $A=B_NB_N^{*}$ (resp. $A=B_N^{*}B_N$) if and only if $A$ is a Fredholm operator. In Section 4 we show that for a given operator $A$ in $B(H)$ there exists an operator $B$ in $AlgN$ satisfying $AA^{*}=BB^{*}$ if and only if the range $r(A)$ of $A$ is equal to the range of some operator in $AlgN$. We also determine the algebraic structure of the set of ranges of operators in $AlgN$. Let $F_r(N)$ be the set of positive operators $A$ for which there exists an operator $B$ in $AlgN $ satisfying $A=BB^{*}$. In Section 5 we obtain information about this set. In particular we discuss the following question: Assume $A$ and $B$ are positive operators such that $A\leq B$ and $A$ belongs to $F_r(N)$. Which further conditions permit us to conclude that $B$ belongs to $F_r(N)$?

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  • 1. Adams, G. T., Froelich, J., McGuire, P. J. and Paulsen, V. I., Analytic reproducing kernels and factorization, Indiana U. Math. J. 43 (1994), 839-856. MR 95k:47027
  • 2. Anoussis, M. and Katsoulis, E. G., Factorisation in nest algebras, Proc. Amer. Math. Soc. 125 (1997), 87-92. MR 97c:47053
  • 3. Arveson, W. B., Analyticity in operator algebras, Amer. J. Math. 89 (1967), 578-642. MR 36:6946
  • 4. Arveson, W. B., Interpolation problems in nest algebras, J. Funct. Anal. 53 (1983), 208-233. MR 52:3979
  • 5. Conway, J. B., A course in functional analysis, Springer-Verlag (1985). MR 86h:46001
  • 6. Davidson, K. R., Nest algebras, Pitman Research Notes in Mathematics Series, 191 (1988). MR 90f:47062
  • 7. Devinatz, A., The factorization of operator valued functions, Ann. Math. 73 (1961) 458-495. MR 23:A3997
  • 8. Dixmier, J., Étude sur les varietés et les opérateurs de Julia, Bull. Soc. Math. France 77 (1949), 11-101. MR 11:369f
  • 9. Douglas, R. G., On factoring positive operator functions, J. Math. Mech. 16 (1966) 119-126. MR 35:782
  • 10. Douglas, R. G., On majorization, factorization and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413-416. MR 34:3315
  • 11. Fillmore, P. A. and Williams, J. P., On operator ranges, Advances in Math. 7 (1971), 254-281. MR 45:2518
  • 12. Foias, C., Invariant para-closed subspaces, Indiana U. Math. J. 21 (1972), 887-906. MR 45:2516
  • 13. Gohberg, I. C. and Krein, M. G., Theory and applications of Voltera operators in Hilbert space, Transl. Math. Monographs, 24 (1970), AMS. MR 41:9041
  • 14. Kadison, R. V. and Ringrose J. R., Fundamentals of the theory of operator algebras, Vol. I, Academic Press, (1983). MR 85j:46099
  • 15. Larson, D. R., Nest algebras and similarity transformations, Ann. Math. 121 (1985), 409-427. MR 86j:47061
  • 16. Lowdenslager, D. B., On factoring matrix valued functions, Ann. Math. 78 (1963), 450-454. MR 27:5094
  • 17. McAsey, M., Muhly, P. and Saito, K-S., Nonselfadjoint crossed products (invariant subspaces and maximality), Trans. Amer. Math. Soc. 248 (1979) 381-410. MR 80j:46101b
  • 18. Muhly, P., The function-algebraic ramifications of Wiener's work on prediction theory and random analysis, in: Norbert Wiener: Collected Works with Commentaries, Vol. III, The MIT Press, Cambridge, Mass., (1981) 339-370. MR 83i:01089
  • 19. Pitts, D. R., Factorization problems for nests: Factorization methods and caracterizations of the universal factorization property, J. Funct. Anal. 79 (1988), 57-90. MR 90a:46160
  • 20. Power, S. C., Factorization in analytic operator algebras, J. Funct. Anal. 67 (1986), 413-432. MR 87k:47040
  • 21. Power, S. C., Spectral characterization of the Wold-Zasuhin decomposition and prediction-error operator, Math. Proc. Camb. Phil. Soc. 110 (1991), 559-567. MR 92j:47032
  • 22. Ringrose, J. R., On some algebras of operators, Proc. London Math. Soc. (3) 15 (1965), 61-83. MR 30:1405
  • 23. Shields, A. L, An analogue of a Hardy-Littlewood-Fejer inequality for upper triangular trace class operators, Math. Z. 182 (1983), 473-484. MR 85c:47022

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Additional Information

M. Anoussis
Affiliation: Department of Mathematics, University of the Aegean, Karlovassi, Samos 83200 Greece

E. G. Katsoulis
Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858

Received by editor(s): January 24, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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