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Wave limits and generalized Hilbert transforms

Abstract (Summary)
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Let [...][subscript i] = L[subscript 2]([...],[...]; N[subscript i], where N[subscript i] is a Hilbert space (i = 1.2). Define the operator L by Lf(x) = xf(x), and let [...][subscript I] be the characteristic function of I. We examine bounded linear operators T:[...][subscript 1][...][subscript 2] which satisfy some or all of the following conditions: (1) There exists a complex-valued function K[subscript fg](x,y) on [...][subscript 1] x [...][subscript 2] x R[superscript 2] such that K[subscript fg] [...] L[subscript 1] (I x J), and (T[...][subscript I]f,[...][subscript J]g) = [...][subscript I x [subscript J]K[subscript fg] for disjoint compact intervals I and J. (2) (T[subscript 0]f,g) :[...] K[subsript fg][...] exists for f[epsilon] [...][subscript 1] and g[epsilon] [...][subscript 2]. {X[subscript epsilon] is a suitably chosen family of subregions of {(x,y):x[...]y}. (3) [...] exists. We show that if T satisfies 1 and 2, then [...][subscript Z] (T-T[subscript 0])[...][subscript Z] is a multiplication operator for every bounded interval Z. Then T will satisfy 3 if T[subscript 0] satisfies 3. We also obtain a representation for the limit 3. In case N[subscript 1]=N [subscript 2] = complex numbers, and K(x,y) is the Fourier transform of an integrable function, then T defined by (Tf,g) = [...] satisfies 1, 2 and 3. The theory is applied to the situation V =symmetric operator, H = self-adjoint extension of L+V, and H[subscript 0] = L in the space [...][subscript 1]. Conditions analogous to 1, 2 and 3 are: (1') Replace (T[...][subscript I]f,[...][subscript J]g) in 1 by (E(I)f,E[subscript 0](J)g). (2') The same as 2. (3') [...] exists. We show 1' is satisfied when V is a special Carleman operator, and 1', 2', 3' are satisfied when V is of trace class.
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