It is well known that the multiplication operators [1,2] possess a very rich structure theory, such that these operators played an important role in the study of operators on Hilbert… Click to show full abstract
It is well known that the multiplication operators [1,2] possess a very rich structure theory, such that these operators played an important role in the study of operators on Hilbert Spaces. In this paper, we introduce a multiplication operation that allows us to give to the Carleman integral operator of second class [3,8] the form of a multiplication operator. In what follows, we shall assume that the reader is familiar with the fundamental results and the standard notation of the Integral operators theory [8–12]. Let X be an arbitrary set, μ a σ−finite measure on X (μ is defined on a σ−algebra of subsets of X, we don’t indicate this σ−algebra), L2 (X,μ) the Hilbert space of square integrable functions with respect to μ. Instead of writing “μ−measurable”, “μ−almost everywhere” and “(dμ (x))” we write “measurable”, “a.e.” and “dx”. A linear operator A : D (A) −→ L2 (X,μ), where the domain D (A) is a dense linear manifold in L2 (X,μ), is said to be integral if there exists a measurable function K on X ×X, a kernel, such that, for every f ∈ D (A),
               
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