arXiv subjects
Publications and source records attributed to Miguel Lacruz.
It is shown that if the Deddens algebra ${\mathcal D}_T$ associated with a quasinilpotent operator $T$ on a complex Banach space is closed and localizing then $T$ has a nontrivial closed hyperinvariant subspace.
A complex scalar $λ$ is said to be an extended eigenvalue of a bounded linear operator $T$ on a complex Banach space if there is a nonzero operator $X$ such that $TX= λXT.$ Such an operator $X$ is called an extended eigenoperator of $T$ corresponding to the extended eigenvalue $λ.$ The purpose of this paper is to give a description of the extended eigenvalues for the discrete Cesàro operator $C_0,$ the finite continuous Cesàro operator $C_1$ and the infinite continuous Cesàro operator $C_\infty$ defined on the complex Banach spaces $\ell^p,$ $L^p[0,1]$ and $L^p[0,\infty)$ for $1 < p <\infty$ by the expressions \begin{align*} (C_0f)(n) \colon & = \frac{1}{n+1} \sum_{k=0}^n f(k),\\ (C_1f)(x) \colon & = \frac{1}{x} \int_0^x f(t)\,dt,\\ (C_\infty f)(x) \colon & = \frac{1}{x} \int_0^x f(t)\,dt. \end{align*} It is shown that the set of extended eigenvalues for $C_0$ is the interval $[1,\infty),$ for $C_1$ it is the interval $(0,1],$ and for $C_\infty$ it reduces to the singleton $\{1\}.$
It is shown that the algebra \(L^\infty(μ)\) of all bounded measurable functions with respect to a finite measure \(μ\) is localizing on the Hilbert space \(L^2(μ)\) if and only if the measure \(μ\) has an atom. Next, it is shown that the algebra \(H^\infty({\mathbb D})\) of all bounded analytic multipliers on the unit disc fails to be localizing, both on the Bergman space \(A^2({\mathbb D})\) and on the Hardy space \(H^2({\mathbb D}).\) Then, several conditions are provided for the algebra generated by a diagonal operator on a Hilbert space to be localizing. Finally, a theorem is provided about the existence of hyperinvariant subspaces for operators with a localizing subspace of extended eigenoperators. This theorem extends and unifies some previously known results of Scott Brown and Kim, Moore and Pearcy, and Lomonosov, Radjavi and Troitsky.
We survey several applications of fixed point theorems in the theory of invariant subspaces. The general idea is that a fixed point theorem applied to a suitable map yields the existence of invariant subspaces for an operator on a Banach space.
An algebra of bounded linear operators on a Banach space is said to be {\em strongly compact} if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be {\em strongly compact} if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We provide a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fernández-Valles and the first author. Further applications are also derived, for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator.
We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator.