In mathematics , a Neumann series (or Neumann series ) is a series of the form , where is a continuous linear operator on a normalized space and .
${\ displaystyle \ textstyle \ sum _ {n = 0} ^ {\ infty} T ^ {n}}$${\ displaystyle T: X \ rightarrow X}$${\ displaystyle \ left.X \ right.}$${\ displaystyle \ left.T \ right. ^ {0}: = \ mathrm {Id} _ {X}}$

Be a normed space and a compact operator, . It is , thus steady - - the space of linear bounded operators on .
${\ displaystyle \ left (X, \ left \ | \ cdot \ right \ | \ right)}$${\ displaystyle T \ colon X \ rightarrow X}$${\ displaystyle T \ in L (X)}$${\ displaystyle L (X)}$${\ displaystyle X}$

If the Neumann series converges in space with respect to the operator norm , then is invertible and it holds${\ displaystyle \ sum _ {n = 0} ^ {\ infty} T ^ {n}}$${\ displaystyle \ left.L (X) \ right.}$${\ displaystyle A = \ left (\ mathrm {Id} -T \ right)}$

${\ displaystyle A ^ {- 1} = \ left (\ mathrm {Id} -T \ right) ^ {- 1} = \ sum \ limits _ {k = 0} ^ {\ infty} T ^ {k}}$.

The Neumann series converges if is a Banach space and holds for the operator norm . Then also applies:${\ displaystyle \ left (X, \ left \ |. \ right \ | \ right)}$${\ displaystyle \ left \ | T \ right \ | <1}$

${\ displaystyle \ left \ | (\ mathrm {Id} -T) ^ {- 1} \ right \ | \ leq \ left (1- \ | T \ | \ right) ^ {- 1}}$.

Weaker conditions are also known under which the series can converge, e.g. B. it is sufficient if the condition is only valid for a power of the operator . Then${\ displaystyle T}$${\ displaystyle \; \ | T ^ {n} \ | <1 \;}$

${\ displaystyle {\ begin {aligned} (IT) ^ {- 1} = & \ left (I + T + T ^ {2} + \ dots + T ^ {n-1} \ right) \ cdot \ left ( IT ^ {n} \ right) ^ {- 1} \\ = & \ left (I + T + T ^ {2} + \ dots + T ^ {n-1} \ right) \ cdot \ sum \ limits _ {k = 0} ^ {\ infty} T ^ {kn}. \ end {aligned}}}$

Invertibility of linear operators

Is a Banach space, e.g. B. , and a bounded operator, e.g. B. a square matrix , then for each scaling factor as
${\ displaystyle V}$${\ displaystyle V = \ mathbb {R} ^ {n}}$${\ displaystyle A \ colon V \ to V}$${\ displaystyle A \ in \ mathbb {R} ^ {n \ times n}}$${\ displaystyle A}$${\ displaystyle \ gamma> 0}$

${\ displaystyle A = {\ tfrac {1} {\ gamma}} (I-T _ {\ gamma}) \;}$ With ${\ displaystyle \; T _ {\ gamma} = I- \ gamma \, A}$

being represented.

If there is now a scaling factor with which the induced operator norm applies, then it is invertible and the inverse is, using the Neumann series,
${\ displaystyle \ | T _ {\ gamma} \ | _ {{} _ {V \ to V}} <1}$${\ displaystyle A}$

${\ displaystyle A ^ {- 1} = \ gamma \ left (I + \ sum _ {k = 1} ^ {\ infty} T _ {\ gamma} {} ^ {k} \ right) = \ gamma \ left (I + \ sum _ {k = 1} ^ {\ infty} (I- \ gamma \, A) ^ {k} \ right)}$.

Openness of the set of invertible operators

Let be two Banach spaces and an invertible operator. Then for every further operator :
${\ displaystyle B, B '}$${\ displaystyle S \ colon B \ to B '}$${\ displaystyle T \ colon B \ to B '}$

If the estimate with applies to the distance in the operator norm from to , then it is also invertible and the inverse has the operator norm
${\ displaystyle S}$${\ displaystyle T}$${\ displaystyle \ | TS \ | \ leq q \, \ | S ^ {- 1} \ | ^ {- 1}}$${\ displaystyle 0 <q <1}$${\ displaystyle T}$

As proof: It is broken down and the Neumann series is applied to the second factor. Convergence is assured, because according to the prerequisite:
${\ displaystyle T = S (I- (IS ^ {- 1} T))}$

${\ displaystyle \ | IS ^ {- 1} T \ | \ leq \ | S ^ {- 1} \ | \, \ | ST \ | \ leq q <1}$.

As a result, the set of invertible operators is open with regard to the topology of the operator norm.