Abstract
LetVbe an arbitrary system of weights on an open connected subsetGofℂN(N≥1)and letB(E)be the Banach algebra of all bounded linear operators on a Banach spaceE. LetHVb(G,E)andHV0(G,E)be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functionsH(G)to the weighted spaces of analytic functionsHVb(G,E)andHV0(G,E).
Highlights
Multiplication operators and composition operators, on different spaces of analytic functions, have been actively appearing in different areas of mathematical sciences like dynamical systems, theory of semigroups, isometries, and, in turn, the theory of weighted composition operators besides their role in the theory of operator algebras and operator spaces
Many authors like Attele [12], Axler [13,14,15,16], Bercovici [17], Eschmeier [18], Luecking [19], Vukotic [20], and Zhu [21] have made a study of multiplication operators on Bergman spaces, whereas Campbell and Leach [22], Feldman [23], Lin [10], and Ohno and Takagi [24] have obtained a study of these operators on Hardy spaces
Besides these well-known analytic function spaces, a study of these operators on some other Banach spaces of analytic functions has been pursued by Bonet et al [31,32,33,34], Contreras and Hernandez-Dıaz [35], Ohno and Takagi [24], and Shields and Williams [36, 37]
Summary
Multiplication operators ( known as multipliers) and composition operators, on different spaces of analytic functions, have been actively appearing in different areas of mathematical sciences like dynamical systems, theory of semigroups, isometries, and, in turn, the theory of weighted composition operators besides their role in the theory of operator algebras and operator spaces. Evard and Jafari [1] and Siskakis [2, 3] have employed these operators to make a study of weighted composition semigroups and dynamical systems on Hardy Spaces. De Leeuw et al [4] and Nagasawa [5] have described isometries of Hardy spaces H1(D) and H∞(D) as a product of multiplication operators and composition operators. In [11], Arveson has recently obtained Toeplitz C∗-algebras and operator spaces associated with these multiplication operators on Hardy Spaces
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