Citation
Derzko, Nicholas Anthony (1965) Wave limits and generalized Hilbert transforms. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd04022003092750
Abstract
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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 complexvalued 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] (TT[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 = selfadjoint 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.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  1 April 1965 
Record Number:  CaltechETD:etd04022003092750 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd04022003092750 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  1245 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  07 Apr 2003 
Last Modified:  26 Dec 2012 02:36 
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