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

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 $A_k={⨁}_{j=1}^n{\lambda }_j{J}_k^{\alpha }$ defined on ${⨁}_{j=1}^n{W}_p^k[0,1]$. Namely, we describe the commutant $\{A_k{\}}^{\prime }$, the double commutant $\{A_k{\}}^{\prime \prime }$ 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.

published in Integr. equ. oper. theory 63 (2009), 181-215

References in corpus (1)

• Intertwining relations and extended eigenvalues for analytic Toeplitz operators