One of the useful features of spectral measures which happen to be equicontinuous is that their associated integration maps are bicontinuous isomorphisms of the corresponding L1-space onto their ranges. It is shown here that equicontinuity is not necessary for this to be the case; a somewhat weaker property suffices. This is of some interest in practice since there are many natural examples of spectral measures which fail to be equiconontinuous.

Let $\text{Alg}\left( \mathcal{L} \right)$ be the algebra of all bounded linear operators on a normed linear space $\mathcal{X}$ leaving invariant each member of the complete lattice of closed subspaces $\mathcal{L}$. We discuss when the subalgebra of finite rank operators in $\text{Alg}\left( \mathcal{L} \right)$ is non-zero, and give an example which shows this subalgebra may be zero even for finite lattices. We then give a necessary and sufficient lattice condition for decomposing a finite rank operator $F$ into a sum of a rank one operator and an operator whose range is smaller than that of $F$, each of which lies in $\text{Alg}\left( \mathcal{L} \right)$. This unifies results of Erdos, Longstaff, Lambrou, and Spanoudakis. Finally, we use the existence of finite rank operators in certain algebras to characterize the spectra of Riesz operators (generalizing results of Ringrose and Clauss) and compute the Jacobson radical for closed algebras of Riesz operators and $\text{Alg}\left( \mathcal{L} \right)$ for various types of lattices.

In this note we prove that one aspect of the similarity theory, for the Volterra nest in Lp(0,1 ) for 1 < p ≠ 2 < ∞, is like that for p = 1 ; we thus answer a question from [ALWW].

Let H be a Hilbert space, dim H ≥ 3, and B(H) the algebra of all bounded linear operators on H. We characterize bijective linear mappings on B(H) that preserve normal operators.

It is proved that linear mappings of matrix algebras which preserve idempotents are Jordan homomorphisms. Applying this theorem we get some results concerning local derivations and local automorphisms. As an another application, the complete description of all weakly continuous linear surjective mappings on standard operator algebras which preserve projections is obtained. We also study local ring derivations on commutative semisimple Banach algebras.

Let X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.