Subnormal Operators, Dilation Theory And The Classes A(,n) (dual Algebras).

Let H be a complex infinite-dimensional separable Hilbert space and L(H) be the algebra of all bounded operators on H. If T in L(H) is an absolutely continuous contraction one can define f(T) for f belonging to H('(INFIN)) using the Sz.-Nagy - Foias functional calculus. We say that T belongs to if (VBAR)(VBAR)f(T)(VBAR)(VBAR) = sup (VBAR)f((lamda))(VBAR) : (VBAR)(lamda)(VBAR) < 1 for all f in H('(INFIN)). In this dissertation we give necessary and sufficient conditions for a subnormal operator to belong to . If T is any operator in L(H) then A(,T) is the smallest unital sub- algebra of L(H) containing T and closed in the ultraweak operator topology. If 1 (LESSTHEQ) n (LESSTHEQ) (ALEPH)(,0), then T belongs to (,n) if T belongs to and given (phi)(,ij) : 0 (LESSTHEQ) i,j < n , a collection of ultraweakly continuous linear functionals on A(,T), one can find sequences x(,i) : 0 (LESSTHEQ) i < n and y(,j) : 0 (LESSTHEQ) j < n contained in H such that (phi)(,ij) (A) = (Ax(,i),y(,j)) for every A in A(,T)- and for 0 (LESSTHEQ) i,j < n. This dissertation gives sufficient conditions for a normal operator to belong to (,n). There is an extensive dilation theory for operators belonging to (,(ALEPH)(,0)). This dissertation shows that this dilation theory can be extended to subnormal operators S for which the left-essential spectrum of S intersected with the open unit disc is sufficiently rich. These subnormal operators need not be contractions.Ph.D.MathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/127931/2/8621385.pd