Telescoping bound for products of operators

If

A_{j}

and

B_{j}

have operator norm at most one, then inserting and subtracting one factor at a time gives

\prod_{j = 1}^{m} A_{j} - \prod_{j = 1}^{m} B_{j} \leq \sum_{j = 1}^{m} ∥ A_{j} - B_{j} ∥.

Table of contents
- Unitary product telescoping Telescoping bound for products of operators

Unitary product telescoping ( $∥ U_{m} \dots U_{1} - V_{m} \dots V_{1} ∥ \leq \sum_{j} ∥ U_{j} - V_{j} ∥$ )

Insert one factor difference at a time: the

j

th term is

U_{m} \dots U_{j + 1} (U_{j} - V_{j}) V_{j - 1} \dots V_{1}

. Multiplication by the surrounding unitary operators preserves the spectral norm, so the triangle inequality gives the displayed dimension-independent bound. No commutation assumption is required.

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