Power series

We consider next the so-called Neumann series , where is a linear transformation with norm on a finite-dimensional vector space. If we write then To prove that has a limit as , we consider (for any two indices and with ) Since , the last written quantity approaches zero as ; it follows that has a limit as . To evaluate the limit we observe that is invertible. (Proof: implies that , and, if , this implies that , a contradiction.) Hence we may write (1) in the form since as , it follows that .

As another example of an infinite series of transformations we consider the exponential series. For an arbitrary linear transformation (not necessarily with ) we write Since we have and since the right side of this inequality, being a part of the power series for , converges to as , we see that there is a linear transformation such that . We write ; we shall merely mention some of the elementary properties of this function of .

Consideration of the triangular forms of and of shows that the proper values of , together with their algebraic multiplicities, are equal to the exponentials of the proper values of . (This argument, as well as some of the ones that follow, applies directly to the complex case only; the real case has to be deduced via complexification.) From the consideration of the triangular form it follows also that the determinant of , that is, , where are the (not necessarily distinct) proper values of , is the same as . Since , this shows, incidentally, that is always invertible.

Considered as a function of linear transformations the exponential retains many of the simple properties of the ordinary numerical exponential function. Let us, for example, take any two commutative linear transformations and . Since is the limit (as ) of the expression we will have proved the multiplication rule for exponentials when we have proved that this expression converges to zero. (Here stands for the combinatorial coefficient .) An easy verification yields the fact that for the product occurs in both terms of the last written expression with coefficients that differ in sign only. The terms that do not cancel out are all in the subtrahend and are together equal to the summation being extended over those values of and that are and for which . Since implies that at least one of the two integers and is greater than the integer part of (in symbols ), the norm of this remainder is dominated by where and as .

Similar methods serve to treat , where is any function representable by a power series, and where is (strictly) smaller than the radius of convergence of the series. We leave it to the reader to verify that the functional calculus we are here hinting at is consistent with the functional calculus for normal transformations. Thus, for example, as defined above is the same linear transformation as is defined by our previous notion of in case is normal.

EXERCISES