Isometric operator

A mapping $U$ of a metric space $(X,\rho_X)$ into a metric space $(Y,\rho_Y)$ such that

$$\rho_X(x_1,x_2)=\rho_Y(Ux_1,Ux_2)$$

for all $x_1,x_2\in X$. If $X$ and $Y$ are real normed linear spaces, $U(X)=Y$ and $U(0)=0$, then $U$ is a linear operator.

An isometric operator $U$ maps $X$ one-to-one onto $U(X)$, so that the inverse operator $U^{-1}$ exists, and this is also an isometric operator. The conjugate of a linear isometric operator from some normed linear space into another is also isometric. A linear isometric operator mapping $X$ onto the whole of $Y$ is said to be a unitary operator. The condition for a linear operator $U$ acting on a Hilbert space $H$ to be unitary is the equation $U^*=U^{-1}$. The spectrum of a unitary operator (cf. Spectrum of an operator) lies on the unit circle, and $U$ has a representation

$$U=\int\limits_0^{2\pi}e^{i\phi}dE_\phi,$$

where $\{E_\phi\}$ is the corresponding resolution of the identity. An isometric operator defined on a subspace of a Hilbert space and taking values in that space can be extended to a unitary operator if the orthogonal complement of its domain of definition and its range have the same dimension.

With every symmetric operator $A$ with domain of definition $D_A\subset H$ is associated the isometric operator

$$U_A=(A-iI)(A+iI)^{-1},$$

called the Cayley transform of $A$. If $A$ is self-adjoint, then $U_A$ is unitary.

Two operators $A$ and $B$ with the same domain of definition $D$ are said to be metrically equal if $B=UA$, where $U$ is an isometric operator, that is, if $\|Bx\|=\|Ax\|$ for all $x\in D$. Such operators have a number of properties in common. For every bounded linear operator $A$ acting on a Hilbert space there exists one and only one positive operator metrically equal to it, namely that defined by the equality $B=\sqrt{A^\ast A}$.

References

[1]	N.I. Akhiezer, I.M. Glazman, "Theory of linear operators on a Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)
[2]	A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1965) (Translated from Russian)
[3]	B. Mazur, S. Ulam, "Sur les transformations isométriques d'espaces vectoriels normés" C.R. Acad. Sci. Paris , 194 (1932) pp. 946–948