View source for Unitarily-equivalent operators

Linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u0955101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u0955102.png" />, acting in a Hilbert space, with domains of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u0955103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u0955104.png" />, respectively, such that: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u0955105.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u0955106.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u0955107.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u0955108.png" /> is a [[Unitary operator|unitary operator]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u0955109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551010.png" /> are bounded linear operators, then 1) may be omitted. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551011.png" /> is a [[Self-adjoint operator|self-adjoint operator]], then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551012.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551014.png" /> are bounded operators, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551015.png" />.

Self-adjoint unitarily-equivalent operators have unitarily-equivalent spectral functions, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551016.png" />. Therefore the spectra of unitarily-equivalent operators have identical structures: either both are pure point spectra, or both are purely continuous or both are mixed. In particular, in the case of a pure point spectrum the eigenvalues of unitarily-equivalent operators are identical and the multiplicities of corresponding eigenvalues coincide; moreover, this is not only a necessary but also a sufficient condition for the unitary equivalence of operators with a pure point spectrum.

Examples of pairs of unitarily-equivalent operators in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551017.png" /> are the differentiation operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551018.png" />, with domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551019.png" /> consisting of all functions that are absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551020.png" /> and that have a square-summable derivative in this interval, and the operator of multiplication by the independent variable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095510/u09551021.png" />. In this case the unitary operator accomplishing the unitary equivalence is the [[Fourier transform|Fourier transform]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. Akhiezer,   I.M. Glazman,   "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. [L.A. Lyusternik] Liusternik,   "Elements of functional analysis" , F. Ungar  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Riesz,   B. Szökefalvi-Nagy,   "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>



====Comments====
For non-self-adjoint operators, characteristic operator-functions provide a tool to identify classes of unitarily-equivalent operators. See [[#References|[3]]] and [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.S. Brodskii,   "Triangular and Jordan representations of linear operators" , Amer. Math. Soc.  (1971)  (Translated from Russian)</TD></TR></table>