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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),

Carleman integral operators as multiplication operators and perturbation theory