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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" />.
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Linear operators $A$ and $B$, acting in a Hilbert space, with domains of definition $D_A$ and $D_B$, respectively, such that: 1) $UD_A=D_B$; and 2) $UAU^{-1}x=Bx$ for any $x\in D_B$, where $U$ is a [[Unitary operator|unitary operator]]. If $A$ and $B$ are bounded linear operators, then 1) may be omitted. If $A$ is a [[Self-adjoint operator|self-adjoint operator]], then so is $B$; if $A$ and $B$ are bounded operators, then $\|A\|=\|B\|$.
  
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.
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Self-adjoint unitarily-equivalent operators have unitarily-equivalent spectral functions, i.e. $E_\lambda(B)=UE_\lambda(A)U^{-1}$. 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]].
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Examples of pairs of unitarily-equivalent operators in the complex space $L_2(-\infty,\infty)$ are the differentiation operator $Ax=idx/dt$, with domain of definition $D_A$ consisting of all functions that are absolutely continuous on $(-\infty,\infty)$ and that have a square-summable derivative in this interval, and the operator of multiplication by the independent variable, $Bx=tx(t)$. In this case the unitary operator accomplishing the unitary equivalence is the [[Fourier transform|Fourier transform]].
  
 
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Latest revision as of 11:04, 22 August 2014

Linear operators $A$ and $B$, acting in a Hilbert space, with domains of definition $D_A$ and $D_B$, respectively, such that: 1) $UD_A=D_B$; and 2) $UAU^{-1}x=Bx$ for any $x\in D_B$, where $U$ is a unitary operator. If $A$ and $B$ are bounded linear operators, then 1) may be omitted. If $A$ is a self-adjoint operator, then so is $B$; if $A$ and $B$ are bounded operators, then $\|A\|=\|B\|$.

Self-adjoint unitarily-equivalent operators have unitarily-equivalent spectral functions, i.e. $E_\lambda(B)=UE_\lambda(A)U^{-1}$. 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 $L_2(-\infty,\infty)$ are the differentiation operator $Ax=idx/dt$, with domain of definition $D_A$ consisting of all functions that are absolutely continuous on $(-\infty,\infty)$ and that have a square-summable derivative in this interval, and the operator of multiplication by the independent variable, $Bx=tx(t)$. In this case the unitary operator accomplishing the unitary equivalence is the Fourier transform.

References

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


Comments

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

References

[a1] M.S. Brodskii, "Triangular and Jordan representations of linear operators" , Amer. Math. Soc. (1971) (Translated from Russian)
How to Cite This Entry:
Unitarily-equivalent operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitarily-equivalent_operators&oldid=11796
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article