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

Abstract

Let J α k be a real power of the integration operator J k defined on the Sobolev space W k p [0, 1]. We investigate the spectral properties of the operator A k = n j=1 λjJ α k defined on n j=1 W k p [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 µA k of A k and describe the set Cyc A k of its cyclic subspaces. In passing, we present a simple counterexample for the implicationto be valid. (2000). Primary 47A15, 47A16, 47L80; Secondary 47L10. Mathematics Subject Classification

Some applications of Banach algebra techniques

“…Remark 3.8 By applying the methods of the papers 3, 16 and [4, Proposition 4.6] it can be proved the same results, as in Theorem 12, for the operators V k , \documentclass{article}\usepackage{amssymb,amsmath}\begin{document}\pagestyle{empty}$k\in \mathbb {N},$\end{document} in the space C ( n ) [0, 1], which is left for the reader.…”

“…(b) Let λ > 1 and BV = λ VB . Following by Domanov and Malamud (see the proof of Proposition 4.6 in 4), consider the block matrix representations of the operators V and B with respect to the direct sum decomposition C ( n ) [0, 1] = C ( n ) n , 0 [0, 1]∔ X n , where X n ≔ span {1, x , …, x n }. Since C ( n ) n , 0 [0, 1] ∈ Lat V , one has Further, a slight modification of the proof of the assertion (2.3) in Proposition 4.6 of the paper 4 (where the same questions are considered for the operator V α , α > 0, on the Sobolev space W k p [0, 1]) gives the proof of (b) (which is omitted).…”

“…Recall that to the study of extended eigenvalues and extended eigenvectors are devoted, for example, the works of Malamud, Domanov, Biswas, Lambert, Petrovic (for more information see, for instance, the recent paper by Domanov and Malamud 4 and references therein). In Section 4, we characterize in terms of the notion of extended eigenvector some special classes of composition operators ( C θ f )( x ) = f (θ( x )) on the Lebesgue spaces L p = L p [0, 1] (1 ≤ p < ∞).…”

On some operator equations in the space of analytic functions and related questions

“…The set of intertwining operators for the pair {V^, AV^} with ß > 0 and X eC was studied by Malamud in [3,9,10]. Namely, he showed that there exists a nonzero intertwining operator for the pair |V^, AV^ } only if A > 0.…”

Some concrete operators and their properties

“…Our main method for the proofs of the obtained results is the Duhamel products method, which was essentially used in investigation of various questions of analysis, including differential and integrodifferential equations, the boundary value problems of mathematical physics, operator theory, and Banach algebras in the works of Nagnibida [27], Fage and Nagnibida [11], Dimovski [9], Tkachenko [34], Malamud [25,26], Domanov and Malamud [10], Bojinov [4], Wigley [36,37], Karaev [17], and Karaev et al [19].…”