Problem. If AB=BA, show that r(A+B)≤r(A)+r(B).This problem is stated in a problem book and described as an easy consequence of the definition of r(A) without proof, we work out the detail:
Proof.
Since r(cA)=cr(A) for any c≥0, we can assume that ‖A‖,‖B‖<1. Now there is an N such that for every k,n such that k≥N and n−k≥N,
‖An−k‖≤(r(A)+ϵ)n−kand‖Bk‖≤(r(B)+ϵ)k. Here ϵ can be arbitrarily small such that both r(A)+ϵ,r(B)+ϵ<1.
Now we multiply them and take summation to get
‖∑k≥Nn−k≥N(nk)An−kBk‖≤n−N∑k=N(nk)(r(A)+ϵ)n−k(r(B)+ϵ)k and if we add both sides by what's missing on the left in (A+B)n and also add the remaining terms of the series on the RHS, we have
‖(A+B)n‖≤‖∑k>n−N, ork<N(nk)An−kBk‖+(r(A)+r(B)+2ϵ)n. Now let's split the summation into two parts, the RHS above
≤∑k>n−N(nk)‖An−k‖‖Bk‖+∑k<N(nk)‖An−k‖‖Bk‖+(r(A)+r(B)+2ϵ)n≤∑k>n−N(nk)⋅1⋅(r(B)+ϵ)k+∑k<N(nk)⋅(r(A)+ϵ)n−k⋅1+(r(A)+r(B)+2ϵ)n≤N(nN)(r(B)+ϵ)n−N+N(nN)(r(A)+ϵ)n−N+(r(A)+r(B)+2ϵ)n≤2N(nN)(r(A)+r(B)+2ϵ)n−N+(r(A)+r(B)+2ϵ)n=(2N(nN)(r(A)+r(B)+2ϵ)−N+1)(r(A)+r(B)+2ϵ)n, we assume that in (∗) n is is big enough to make sure n−N>N. Now we take nth root and take lim on both sides ((nN) is just a scalar times finite many linear factors in n) to get
r(A+B)≤r(A)+r(B)+2ϵ.
Finally set ϵ→0+.◼
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