Boundedness criteria for Boolean algebras of projections
For a Banach space \(X\), it is not always the case that the sum or product of two commuting spectral operators is also spectral. This is related to the Boolean algebra of projections generated by a pair of commuting bounded Boolean algebras of projections being bounded. Though results were known for \(L^p\) spaces, the author provides more general geometric conditions on \(X\) for the generated Boolean algebra of projections to be bounded. Let \(E\) and \(F\) be two commuting bounded Boolean algebras of projections on \(X\). It is shown that if \(X\) is a Banach lattice or a subspace of a \(p\)-concave Banach lattice, then the Boolean algebra generated by \(E\) and \(F\) is bounded. \(X\) is \(p\)-concave if \[ (\sum\| x_i\|^p)^{1/p}\leq M\|(\sum| x_i|^p)^{1/p}\| \] for each finite sequence in \(X\). The results are then applied to the above question of establishing if the sum or product of commuting spectral operators are spectral operators. A further application to Banach algebras of the type \(C(K)\) is considered.
sum or product of two commuting spectral operators
scientific article; zbMATH DE number 1054123
scientific article