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Abstract. We provide an example of an elementary operator which leaves invariant a nest algebra but which cannot be written as a finite sum of multiplications each of which leaves the nest algebra invariant. We also prove that the given operator lies in the completely bounded norm closure of the linear span of the multiplications which leave the nest algebra invariant.Let A be a Banach algebra and let A, B ∈ A. A multiplication (operator) A · B :The length l of R is defined to be the smallest number of multiplication terms required for any representation of R.Elementary operators Let C be a subalgebra of A. David Larson has suggested the study of E C (A), the algebra of elementary operators on A which leave C invariant. With A = B(l 2 ) and C = T ∞ , the nest algebra of upper triangular operators with respect to the standard basis in l 2 , we provide in Theorem 3 an example of an R ∈ E C (A) which has no representation m t=1 C t · D t such that C t ·D t ∈ E C (A), t = 1, 2, . . . , m. (This example was discussed, without proof, in [Co].) We also demonstrate in Proposition 4 that R lies in the completely bounded norm closed linear span of the length one elements of E C (A).T ∞ is a commutative subspace lattice (CSL) algebra. Recall that a CSL algebra is a reflexive algebra whose lattice of invariant subspaces is commutative. For finite dimensional CSL algebras there is no operator like the operator R of the previous paragraph.Proof. Let R = p s=1 A s · B s . Since C is a CSL algebra, it admits a star diagram, that is, there exists a subset I × J ⊆ {1, 2, . . . , n} × {1, 2, . . . , n} such that C = (a ij ) : a ij ∈ C, (i, j) ∈ I × J [KL].
Abstract. We provide an example of an elementary operator which leaves invariant a nest algebra but which cannot be written as a finite sum of multiplications each of which leaves the nest algebra invariant. We also prove that the given operator lies in the completely bounded norm closure of the linear span of the multiplications which leave the nest algebra invariant.Let A be a Banach algebra and let A, B ∈ A. A multiplication (operator) A · B :The length l of R is defined to be the smallest number of multiplication terms required for any representation of R.Elementary operators Let C be a subalgebra of A. David Larson has suggested the study of E C (A), the algebra of elementary operators on A which leave C invariant. With A = B(l 2 ) and C = T ∞ , the nest algebra of upper triangular operators with respect to the standard basis in l 2 , we provide in Theorem 3 an example of an R ∈ E C (A) which has no representation m t=1 C t · D t such that C t ·D t ∈ E C (A), t = 1, 2, . . . , m. (This example was discussed, without proof, in [Co].) We also demonstrate in Proposition 4 that R lies in the completely bounded norm closed linear span of the length one elements of E C (A).T ∞ is a commutative subspace lattice (CSL) algebra. Recall that a CSL algebra is a reflexive algebra whose lattice of invariant subspaces is commutative. For finite dimensional CSL algebras there is no operator like the operator R of the previous paragraph.Proof. Let R = p s=1 A s · B s . Since C is a CSL algebra, it admits a star diagram, that is, there exists a subset I × J ⊆ {1, 2, . . . , n} × {1, 2, . . . , n} such that C = (a ij ) : a ij ∈ C, (i, j) ∈ I × J [KL].
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