= Induced signal gain
{title2=$\|T\|_{\rm ind}$}
An \b[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 $T0=0$,
$$
\|T\|_{\rm ind}=\sup_{u\ne0}\frac{\|Tu\|_{\rm out}}{\|u\|_{\rm in}}.
$$
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.<Linear Algebra Done Right by Sheldon Axler (4th edition)>[§7F, norm characterization7.88, printedp282]{ref}
The norms are part of the definition. An original finite-dimensional example uses
$$
M=\begin{pmatrix}1&1\\0&0\end{pmatrix},\qquad Mu=(u_1+u_2,0).
$$
With the same maximum-component norm at input and output, $\|Mu\|_\infty\leq2\|u\|_\infty$, and $u=(1,1)$ attains the ratio2. With the <Euclidean norm>, the <Cauchy-Schwarz inequality> gives $|u_1+u_2|\leq\sqrt2\|u\|_2$, with equality at $u=(1,1)$. With the sum-of-absolute-components norm, $|u_1+u_2|\leq|u_1|+|u_2|$, and $u=(1,0)$ attains ratio1. The same matrix therefore has induced gains2, $\sqrt2$ 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 $\|T_1u\|\leq\gamma_1\|u\|$ and $\|T_2v\|\leq\gamma_2\|v\|$, then
$$
\|T_2T_1u\|\leq\gamma_2\gamma_1\|u\|.
$$
The 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.
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