Compact multiplication operators on semicrossed products

Let A be a Banach algebra and a, b ∈ A. The map Ma,b : A → A given by Ma,b(x) = axb is called a multiplication operator. Properties of compact multiplication operators have been investigated since 1964 when Vala published his work “On compact sets of compact operators” [15]. Let X be a normed space and B(X ) the algebra of all bounded linear maps from X into X . Vala proved that a nonzero multiplication operator Ma,b : B(X ) → B(X ) is compact if and only if the operators a, b ∈ B(X ) are both compact. Also, in [16] Vala defines an element a of a normed algebra to be compact if the mapping x 7→ axa is compact. This concept enabled the study of compactness properties of elements of abstract normed algebras. Ylinen in [17] studied compact elements for abstract C*-algebras and showed that a is a compact element of a C-algebra A if and only if there exists an isometric ∗-representation π of A on a Hilbert space H such that the operator π(a) is compact. Compactness questions have also been considered in the more general framework of elementary operators. A map Φ : A → A, where A is a Banach algebra, is called elementary if Φ = ∑m i=1 Mai,bi for some ai, bi ∈ A, i = 1, . . . ,m. Fong and Sourour showed that an elementary operator Φ : B(H) → B(H), where B(H) is the algebra of bounded linear operators on a Hilbert space H, is compact if and only if there exist compact operators ci, di ∈ B(H), i = 1, . . . ,m such that Φ = ∑m i=1 Mci,di [5]. This result was expanded by Mathieu on prime C*-algebras [9] and later on general C*-algebras by Timoney [14]. Akemann and Wright [1] characterized the weakly compact multiplication operators on B(H), where H is a Hilbert space. Saksman and Tylli [12, 13] and Johnson and Schechtman [6] studied weak compactness of multiplication operators in a Banach space setting. Moreover, strictly singular multiplication operators are studied by Lindström, Saksman and Tylli [8] and Mathieu and Tradacete [10]. Compactness properties of multiplication operators on nest algebras, a class of non selfadjoint operator algebras, are studied by Andreolas and Anoussis in [2]. In