Generalized open mapping theorems for bilinear maps, with an application to operator algebras

by P.G.Dixon

Proc. Amer. Math. Soc., 104 (1988), 106-110.

1991 Mathematics Subject Classification: primary 46A30; secondary 47D35


Cohen [2] gave an example of a surjective bilinear mapping between Banach spaces which was not open, and Horowitz [3] gave a much simpler example. We build on Horowitz's example to produce a similar result for bilinear mappings such that every element of the target space is a linear combination of n elements of the range. An immediate application is that Bercovici's construction [1] of an operator algebra with property (A1) but not (A1(r)) can be extended to achieve property (A1/n) without (A1/n(r)).


  1. H. Bercovici, `Note on property (A1)', Linear Algebra Appl., 91 (1987), 213-216.
  2. P. J. Cohen, `A counterexample to the closed graph theorem for bilinear maps', J. Functional Analysis, 16 (1974), 235-239.
  3. C. Horowitz, `An elementary counterexample to the open mapping principle for bilinear maps', Proc. Amer. Math. Soc., 53 (1975), 293-294.


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