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In the previous chapter we have studied the differential operator ∂ t and its adjoint ∂*t. In this chapter we will study operators involving higher order differential operators and their adjoints, in particular, the various Laplacian operators. These Laplacian operators are closely connected with the Fourier transform, which will be discussed in Chapter 9. With different observations, several infinite dimensional Laplacian operators have been introduced. For example, when we consider the gradient ∇ and its adjoint ∇* with respect to the Gaussian measure µ, then we obtain the number operator N=∇*∇, which we have studied in Chapters 3 and 4. On the other hand, when we consider the Laplacian operator ∆ on ℝn as given by ∆ = ∂ 2/∂x1 2 +...+ ∂ 2/∂xn 2, then the natural generalization is the Gross Laplacian ∆G in an abstract Wiener space, see Gross (1967) and Kuo (1975). But if we take the average of second partial derivatives, i.e. n−1(∂ 2/∂x1 2 + ...+ ∂ 2/∂xn 2), then as n→∞ we get the Lévy Laplacian ∆L. The Lévy Laplacian vanishes, as one can easily guess, on the space (L2). But it plays an important role in harmonic analysis. A related Laplacian, i.e. the Volterra Laplacian ∆V, is a generalization of the Gross Laplacian to the space of generalized white noise functionals. These Laplacian operators have been studied by Gross (1967), Hida (1975a, 1985, 1989), Hida and Saitô (1988), Kubo and Takenaka (1982), Kuo (1975, 1986, 1988a, 1988b, 1990a), Kuo, Obata and Saitô (1990), Obata (1988, 1989, 1990), Piech (1975) and Saitô (1987, 1988), among others.
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