View source for Unitarily-equivalent 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. $E_\lambda(B)=UE_\lambda(A)U^{-1}$. 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|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>