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Thursday, February 12, 2015

The Spectral Radius Inequality: r(A+B)r(A)+r(B)

Let A,B be two elements in a Banach Algebra, the spectral radius is defined by r(A):=limnnAn, and it is a standard fact that the limit above exists (therefore r(A) is properly defined). Recently (in office) I am asked to try:
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 c0, we can assume that A,B<1. Now there is an N such that for every k,n such that kN and nkN,
Ank(r(A)+ϵ)nkandBk(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
kNnkN(nk)AnkBknNk=N(nk)(r(A)+ϵ)nk(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)nk>nN, ork<N(nk)AnkBk+(r(A)+r(B)+2ϵ)n. Now let's split the summation into two parts, the RHS above
k>nN(nk)AnkBk+k<N(nk)AnkBk+(r(A)+r(B)+2ϵ)nk>nN(nk)1(r(B)+ϵ)k+k<N(nk)(r(A)+ϵ)nk1+(r(A)+r(B)+2ϵ)nN(nN)(r(B)+ϵ)nN+N(nN)(r(A)+ϵ)nN+(r(A)+r(B)+2ϵ)n2N(nN)(r(A)+r(B)+2ϵ)nN+(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 nN>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 ϵ0+.

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