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Register a new userThe invariant subspace problem for rank one perturbations
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Authors
Adi Tcaciuc
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We show that for any bounded operator $T$ acting on an infinite dimensional Banach space there exists an operator $F$ of rank at most one such that $T+F$ has an invariant subspace of infinite dimension and codimension. We also show that whenever the boundary of the spectrum of $T$ or $T^*$ does not consist entirely of eigenvalues, we can find such rank one perturbations that have arbitrarily small norm. When this spectral condition is not satisfied, we can still find suitable finite rank perturbations of arbitrarily small norm, but not necessarily of rank one.
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We show that for any bounded operator $T$ acting on infinite dimensional, complex Banach space, and for any $varepsilon>0$, there exists an operator $F$ of rank at most one and norm smaller than $varepsilon$ such that $T+F$ has an invariant subspace of infinite dimension and codimension. A version of this result was proved in cite{T19} under additional spectral conditions for $T$ or $T^*$. This solves in full generality the quantitative version of the invariant subspace problem for rank-one perturbations.
We show that a bounded quasinilpotent operator $T$ acting on an infinite dimensional Banach space has an invariant subspace if and only if there exists a rank one operator $F$ and a scalar $alphainmathbb{C}$, $alpha eq 0$, $alpha eq 1$, such that $T+F$ and $T+alpha F$ are also quasinilpotent. We also prove that for any fixed rank-one operator $F$, almost all perturbations $T+alpha F$ have invariant subspaces of infinite dimension and codimension.
By analytic perturbations, we refer to shifts that are finite rank perturbations of the form $M_z + F$, where $M_z$ is the unilateral shift and $F$ is a finite rank operator on the Hardy space over the open unit disc. Here shift refers to the multiplication operator $M_z$ on some analytic reproducing kernel Hilbert space. In this paper, we first isolate a natural class of finite rank operators for which the corresponding perturbations are analytic, and then we present a complete classification of invariant subspaces of those analytic perturbations. We also exhibit some instructive examples and point out several distinctive properties (like cyclicity, essential normality, hyponormality, etc.) of analytic perturbations.
We study resonances generated by rank one perturbations of selfadjoint operators with eigenvalues embedded in the continuous spectrum. Instability of these eigenvalues is analyzed and almost exponential decay for the associated resonant states is exhibited. We show how these results can be applied to Sturm-Liouville operators. Main tools are the Aronszajn-Donoghue theory for rank one perturbations, a reduction process of the resolvent based on Feshbach-Livsic formula, the Fermi golden rule and a careful analysis of the Fourier transform of quasi-Lorentzian functions. We relate these results to sojourn time estimates and spectral concentration phenomena
Let $ast_P$ be a product on $l_{rm{fin}}$ (a space of all finite sequences) associated with a fixed family $(P_n)_{n=0}^{infty}$ of real polynomials on $mathbb{R}$. In this article, using methods from the theory of generalized eigenvector expansion, we investigate moment-type properties of $ast_P$-positive functionals on $l_{rm{fin}}$. If $(P_n)_{n=0}^{infty}$ is a family of the Newton polynomials $P_n(x)=prod_{i=0}^{n-1}(x-i)$ then the corresponding product $star=ast_P$ is an analog of the so-called Kondratiev--Kuna convolution on a Fock space. We get an explicit expression for the product $star$ and establish a connection between $star$-positive functionals on $l_{rm{fin}}$ and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).
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