On the Spectral Analysis of Direct Sums of Riemann-Liouville Operators in Sobolev Spaces of Vector Functions

TL;DR

This paper analyzes the spectral properties, invariant subspaces, and algebraic structures of direct sums of Riemann-Liouville operators on Sobolev spaces, providing detailed descriptions and counterexamples.

Contribution

It offers a comprehensive spectral analysis of direct sums of Riemann-Liouville operators, including their commutants, invariant subspaces, and spectral multiplicities, which was not previously detailed.

Findings

Description of the commutant and double commutant of the operator

Characterization of invariant and hyperinvariant subspace lattices

Calculation of spectral multiplicity and cyclic subspaces

Abstract

Let $J_k^\alpha$ be a real power of the integration operator $J_k$ defined on Sobolev space $W_p^k[0,1]$. We investigate the spectral properties of the operator $A_k=\bigoplus_{j=1}^n \lambda_j J_k^\alpha$ defined on $\bigoplus_{j=1}^n W_p^k[0,1]$. Namely, we describe the commutant $\{A_k\}'$, the double commutant $\{A_k\}''$ and the algebra $\Alg A_k$. Moreover, we describe the lattices $\Lat A_k$ and $\Hyplat A_k$ of invariant and hyperinvariant subspaces of $A_k$, respectively. We also calculate the spectral multiplicity $\mu_{A_k}$ of $A_k$ and describe the set $\Cyc A_k$ of its cyclic subspaces. In passing, we present a simple counterexample for the implication \Hyplat(A\oplus B)=\Hyplat A\oplus \Hyplat B\Rightarrow \Lat(A\oplus B)=\Lat A\oplus \Lat B to be valid.