Unitarily-equivalent operators

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Linear operators and , acting in a Hilbert space, with domains of definition and , respectively, such that: 1) ; and 2) for any , where is a unitary operator. If and are bounded linear operators, then 1) may be omitted. If is a self-adjoint operator, then so is ; if and are bounded operators, then .

Self-adjoint unitarily-equivalent operators have unitarily-equivalent spectral functions, i.e. . 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 are the differentiation operator , with domain of definition consisting of all functions that are absolutely continuous on and that have a square-summable derivative in this interval, and the operator of multiplication by the independent variable, . In this case the unitary operator accomplishing the unitary equivalence is the Fourier transform.


[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)


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


[a1] M.S. Brodskii, "Triangular and Jordan representations of linear operators" , Amer. Math. Soc. (1971) (Translated from Russian)
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Unitarily-equivalent operators. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article