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Zbl 0654.47023
Larson, David R.
Reflexivity, algebraic reflexivity and linear interpolation. (English)
[J] Am. J. Math. 110, No.2, 283-299 (1988). ISSN 0002-9327; ISSN 1080-6377/e

From the introduction: ``This paper was motivated by a discovery that ``most'' finite dimensional subalgebras of B(H), for H an infinite dimensional Hilbert space, are reflexive, with the only obstructions to reflexivity being finite-rank considerations. Proofs do not depend on multiplicative structure, nor on topology, so extend to linear subspaces of transformations in an abstract setting.'' \par A subspace S of the algebra L(V) of linear operators in a vector space V is said to be algebraically reflexive if $ref\sb a(S)=\{T\in L(V);Tx\in Sx,x\in V\}$ coincides with S. If V is topological and $S\subset B(V)$, the continuous operators, S is reflexive if it coincides with $ref\sb{at}(S):=\{T\in B(V);Tx\in \overline{Sx},x\in V\}$. The main result in section 2 says that for S finite dimensional we have $ref\sb a(S)=S+ref\sb a(S\sb F)$ with $S\sb F$ denoting the finite dimensional operators in S. The results of section 2 extend, in section 3, to certain countably (algebraically) dimensional subspaces of continuous operators acting on a Banach space. Note that reflexivity properties can be interpreted as linear interpolation properties since $T\in L(V)$ interpolates S if and only if $T\in ref\sb a(S).$ \par In section 4 and section 5 some consequences are derived. For example, given continuous operators $S\sb 1,...,S\sb n$ on a Banach space, if for every x there exists a nonzero polynomial $P\sb x$ in n (noncommuting) variables such that $P\sb x(S\sb 1,...,S\sb n)x=0$, then there exists a nonzero polynomial P such that $P(S\sb 1,...,S\sb n)=0$.
[M.Gonzales]

MSC 2000:: *47L30 Abstract operator algebras on Hilbert spaces
47L10 Algebras of operators on Banach spaces, etc.
47A15 Invariant subspaces of linear operators
46M35 Abstract interpolation of topological linear spaces

Keywords: algebraically reflexive; linear interpolation properties

Cited in: Zbl 1201.16032 Zbl 1111.47061

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