Laplace transform methods in multivariate spectral theory
Robert F. Anderson · Pacific Journal of Mathematics · 1974
The Laplace transform of the semigroup exp {tA) generated by an operator A gives the resolvent of A. An integral formula is obtained for the Laplace transform of exp (tA + B), where B is another operator which does not commute with A. The new transform has analytic continuation to the same domain as the resolvent, but the analytic continuation is not single-valued.The integral formula is then applied to the joint spectral theory of noncommutative operators.Explicit computations with matrices of degree two illustrate the results.1* Introduction* Any bounded linear operator A on a Banach space generates a semigroup exp (tA), 0 <^ t < oo, and the Laplace transform -Sf(s, A) of this semigroup converges for Re s sufficiently large and equals the resolvent (s -A)~ι-Sf(s, A) therefore has unique analytic continuation to the component containing oo of the resolvent set A.Multivariate problems requiring integration of exp (X UA,) one variable at a time, lead us to consider the Laplace transform £f(s, A, B) of exp(£A + B\ 0 ^ t < oo, where B is a fixed bounded operator.The main result is: