Invariant Subspaces for Generalized Differentiation and Volterra Operators

TL;DR

The paper develops a unifying framework for invariant subspaces of pairs $(D,V)$ with $DV=I$, where $V$ is compact and quasi-nilpotent and $D$ is densely defined and closed with a one-dimensional kernel, by embedding the problem into a de Branges space model. A central analytic model is constructed via a generalized Fourier transform $\mathcal W$ that identifies $\mathcal W H$ with a shifted de Branges space $e^{-i\alpha z}\mathcal H(E)$, turning $V$ into a rank-one perturbation of the backward shift and $D$ into a multiplication-like operator. The authors establish unicellularity results for generalized Volterra right inverses $V_\beta$, show that at most two such right inverses are unicellular, and characterize all invariant subspaces through the de Branges chain, with explicit descriptions of residual subspaces in terms of intervals and perturbations. They also provide a detailed analysis of $D$-invariant subspaces of $C^{\infty}(D)$, spectrum behavior of restrictions, and annihilator structure, connecting these subspaces to Green's operators for canonical systems and Schrödinger operators. The framework yields concrete consequences for operators with removable spectrum and highlights a deep link between invariant-subspace theory and de Branges spaces, canonical systems, and rank-one perturbations. The results advance understanding of invariant subspaces in broad differential-integral operator settings and suggest avenues for uniqueness theorems in Krein–de Branges-type canonical systems.

Abstract

In this paper we provide a far-reaching generalization of the existent results about invariant subspaces of the differentiation operator $D=\frac{\partial}{\partial t}$ on $C^\infty(0,1)$ and the Volterra operator $Vf(t)=\int_0^tf(s)ds$, on $L^2(0,1)$. We use an abstract approach to study invariant subspaces of pairs $D,V$ with $DV=I$, where $V$ is compact and quasi-nilpotent and $D$ is unbounded densely defined and closed on the same Hilbert space. Our results cover many differential operators, like Schrödinger operators and a large class of other canonical systems, as well as the so-called compact self-adjoint operators with removable spectrum recently studied by Baranov and Yakubovich. Our methods are based on a model for such pairs which involves de Branges spaces of entire functions and plays a crucial role in the development. However, a number of difficulties arise from the fact that our abstract operators do not necessarily identify with the usual operators on such spaces, but with rank one perturbations of those, which, in terms of invariant subspaces creates a number of challenging problems.

Theorems & Definitions (71)

• Definition 1
• Theorem 1
• Theorem 2
• Proposition 1
• proof
• Definition 2
• Theorem 3
• Proposition 2
• proof
• Theorem 4