Abstract
A bounded linear transformation of an abstract Hilbert space C into itself will be called an operator on W. A one-parameter family T(e) of operators on C will be called regular on the interval I eJ <p if it can be expressed as a convergent power series in a real parameter e: T(e) = To+eT+E2T2+ , * * the Tk being operators on XC. The weak, strong, and uniform convergence of this series are equivalent [3].1 If Sn(e) is the partial sum To+eT1+ * +ETn, e being real, Sn*(e)=Tj*+eT?+ . . +enTn*. Since a bounded operator has the same norm as its adjoint, ||T*(E)-Sn*(E)1 =1=11T(e)-Sn(e)JJ--*0 as noo. Therefore
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