Some applications of Banach algebra techniques

Abstract

We consider some functional Banach algebras with multiplications as the usual convolution product * and the so-called Duhamel product . We study the structure of generators of the Banach algebrasWe also use the Banach algebra techniques in the calculation of spectral multiplicities and extended eigenvectors of some operators. Moreover, we give in terms of extended eigenvectors a new characterization of a special class of composition operators acting in the Lebesgue space L p [0,1] by the formula (Cϕ f ) (x) = f (ϕ (x)) .

Some concrete operators and their properties

“…Then the following is an immediate corollary of the above Theorem 2.2 and Theorem 4.1 of paper [19].…”

“…Our next result characterizes in terms of extended eigenvectors a special class of the composition operators C φ on L p [0, 1] (1 ≤ p < +∞) (for the related result, see [19]). We set …”

“…Note that an implication ⇒ in Theorem 4.1 of [19] is proved by using a sufficiently deep result of Malamud (see [25,26]).…”

“…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].…”

“…The * -generators of the Banach algebras (C [0, 1] , * ) and ( C (n) [0, 1] , * ) are investigated in [12] and [19], respectively.…”