Operator

In a normed space, an operator is a linear mapping between two normed spaces. More specifically, let $(X,\parallel \cdot {\parallel }_X)$ and $(Y,\parallel \cdot {\parallel }_Y)$ be two normed spaces, an operator $T$ from $X$ to $Y$ is a linear transformation that maps each vector $x$ in $X$ to a vector $T(x)$ in $Y$.

An important property of operators in normed spaces is that they preserve the norm of vectors. Specifically, if $T$ is an operator from $X$ to $Y$, then for any vector $x$ in $X$, we have $\parallel T(x){\parallel }_Y\le \parallel T\parallel \parallel x{\parallel }_X$, where $\parallel T\parallel$ is the operator norm (induced matrix norm) of $T$. The operator norm of $T$ is defined as the supremum of the ratios $\parallel T(x){\parallel }_Y/\parallel x{\parallel }_X$ over all non-zero vectors $x$ in $X$.

Operators play an important role in functional analysis, where they are used to study linear transformations between normed spaces, such as differential operators and integral operators.

When the operator norm is lesser than 1 it can be applied the Banach fixed point theorem, and then 0 is the only fixed point.

Examples:
translation and momentum operators

Important: common eigenvectors