Abstract
We study operators given by series, in particular, operators of the form \(e^B=\sum\limits_{n=0}^{\infty}{B^n}/{n!},\) where \(B\) is an operator acting in a Banach space \(X\). A corresponding example is provided. In our future research, we will use these operators for introducing and studying functions of operators constructed (with the use of the Cauchy integral formula) on the basis of scalar functions and admitting a faster than power growth at infinity.
Highlights
N=0 is an operator acting in a Banach space X
The authors’ papers [3,4,5] are in the same vein. In these papers, based on the Cauchy integral formula, functions of an operator were constructed in terms of natural powers of the operator
To introduce and study functions of an operator built constructed on the basis of scalar functions and admitting the growth at infinity faster than the power function but not faster than the exponential function have, we will need operators of the form eB =
Summary
N=0 where B is an operator on a Banach space X. We will use series of elements of a Banach space X and operator series. The convergence of partial sums of series from X is interpreted as the convergence in the norm of this space. In X, its sum is the operator A with the domain D(A) = x ∈ X : Anx converges and such n=0 ∞. Suppose that a series an converges in X n=0 ∞. Rekant (ii) Suppose that {am,n}∞ m,n=0 ⊂ X and the series am,n converges absolutely. Every m,n=0 rearrangement of this series converges absolutely to the same sum. The proof of statement (i) is almost the same as the proof of Abel’s test for numerical series.
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