View source for Isometric operator

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

<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>

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.

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

<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>

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.

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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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.

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" />.

====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>