Difference between revisions of "Unitarily-equivalent operators"
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| − | Linear operators | + | {{TEX|done}} |
| + | 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. | + | 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 | + | 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]]. |
====References==== | ====References==== | ||
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) |
Unitarily-equivalent operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitarily-equivalent_operators&oldid=33083