Unitarily-equivalent operators

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)