It is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A.~C.~M. van Rooij and W.~H. Schikhof in 1992.

A band is a semigroup of idempotent operators. A nonnegative band $\cls$ in $\clb(\cll^2 (\clx))$ having at least one element of finite rank and with rank $(S) > 1 $ for all $S$ in $\cls$ is known to have a special kind of common invariant subspace which is termed a standard subspace (defined below). Such bands are called decomposable. Decomposability has helped to understand the structure of nonnegative bands with constant finite rank. In this paper, a geometric characterization of maximal, rank-one, indecomposable nonnegative bands is obtained which facilitates the understanding of their geometric structure.

In this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is $\BR$ or $\BC$.

Let $N$ be an integer which is larger than one. In this paper we study invariant subspaces of $L^2 (\mathbb{T}^N)$ under the double commuting condition. A main result is an $N$-dimensional version of the theorem proved by Mandrekar and Nakazi. As an application of this result, we have an $N$-dimensional version of Lax's theorem.

In this note we give a proof of Lomonosov's extension of Burnside's theorem to infinite dimensional Banach spaces.

Let $\Alg(\l)$ be the algebra of all bounded linear operators on a normed linear space $\x$ leaving invariant each member of the complete lattice of closed subspaces $\l$. We discuss when the subalgebra of finite rank operators in $\Alg(\l)$ 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 $\Alg(\l)$. 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 $\Alg(\l)$ for various types of lattices.