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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i0528201.png" /> of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i0528202.png" /> into a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i0528203.png" /> such that
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A mapping $U$ of a metric space $(X,\rho_X)$ into a metric space $(Y,\rho_Y)$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i0528204.png" /></td> </tr></table>
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$$\rho_X(x_1,x_2)=\rho_Y(Ux_1,Ux_2)$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i0528205.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i0528206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i0528207.png" /> are real normed linear spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i0528208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i0528209.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282010.png" /> is a linear operator.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282011.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282012.png" /> one-to-one onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282013.png" />, so that the inverse operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282015.png" /> onto the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282016.png" /> is said to be a unitary operator. The condition for a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282017.png" /> acting on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282018.png" /> to be unitary is the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282019.png" />. The spectrum of a unitary operator (cf. [[Spectrum of an operator|Spectrum of an operator]]) lies on the unit circle, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282020.png" /> has a representation
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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|Spectrum of an operator]]) lies on the unit circle, and $U$ has a representation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282021.png" /></td> </tr></table>
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$$U=\int\limits_0^{2\pi}e^{i\phi}dE_\phi,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282022.png" /> is the corresponding [[Resolution of the identity|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.
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where $\{E_\phi\}$ is the corresponding [[Resolution of the identity|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282023.png" /> with domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282024.png" /> is associated the isometric operator
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With every symmetric operator $A$ with domain of definition $D_A\subset H$ is associated the isometric operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282025.png" /></td> </tr></table>
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$$U_A=(A-iI)(A+iI)^{-1},$$
  
called the [[Cayley transform|Cayley transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282027.png" /> is self-adjoint, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282028.png" /> is unitary.
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called the [[Cayley transform|Cayley transform]] of $A$. If $A$ is self-adjoint, then $U_A$ is unitary.
  
Two operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282030.png" /> with the same domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282031.png" /> are said to be metrically equal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282033.png" /> is an isometric operator, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282034.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282035.png" />. Such operators have a number of properties in common. For every bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282036.png" /> acting on a Hilbert space there exists one and only one positive operator metrically equal to it, namely that defined by the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052820/i05282037.png" />.
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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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. Akhiezer,  I.M. Glazman,  "Theory of linear operators on a Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Plesner,  "Spectral theory of linear operators" , F. Ungar  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Mazur,  S. Ulam,  "Sur les transformations isométriques d'espaces vectoriels normés"  ''C.R. Acad. Sci. Paris'' , '''194'''  (1932)  pp. 946–948</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. Akhiezer,  I.M. Glazman,  "Theory of linear operators on a Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Plesner,  "Spectral theory of linear operators" , F. Ungar  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Mazur,  S. Ulam,  "Sur les transformations isométriques d'espaces vectoriels normés"  ''C.R. Acad. Sci. Paris'' , '''194'''  (1932)  pp. 946–948</TD></TR></table>

Latest revision as of 12:44, 18 August 2014

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
How to Cite This Entry:
Isometric operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isometric_operator&oldid=32992
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article