Induced signal gain ($\Vert T{\Vert }_{\mathrm{ind}}$)

An induced signal gain is the smallest norm-based amplification bound obtained by taking the worst case over admissible inputs. For a zero-initial-state operator $T$ satisfying $T0=0$,For a linear map this is an operator norm between the specified normed spaces. Axler's definition takes the supremum of output norm over inputs of norm at most1, which is equivalent by linear scaling.^[ref]

The norms are part of the definition. An original finite-dimensional example usesWith the same maximum-component norm at input and output, $\Vert Mu{\Vert }_{\infty }\le 2\Vert u{\Vert }_{\infty }$, and $u=(1,1)$ attains the ratio2. With the Euclidean norm, the Cauchy-Schwarz inequality gives $|u_1+u_2|\le \sqrt{2}\Vert u{\Vert }_2$, with equality at $u=(1,1)$. With the sum-of-absolute-components norm, $|u_1+u_2|\le |u_1|+|u_2|$, and $u=(1,0)$ attains ratio1. The same matrix therefore has induced gains2, $\sqrt{2}$ and1 under these three conventions.

For a time-dependent signal, the choice may additionally compare peak amplitudes or integrated squared amplitudes. The example in input-output gain derives a sharp peak-amplitude bound for a scalar dynamic system. A large response to one trial input supplies a lower bound on induced gain; proving a universal upper bound requires controlling every admissible input.

Compatible induced bounds compose. If $\Vert T_{1}u\Vert \le {\gamma }_1\Vert u\Vert$ and $\Vert T_{2}v\Vert \le {\gamma }_2\Vert v\Vert$, thenThe intermediate norms must be the same. This inequality bounds a cascade without requiring the inputs that maximize the individual blocks to be simultaneously realizable, so the product need not be sharp. In robust performance, both the signal norms and the allowed control model uncertainty must be stated before such a bound can be interpreted.