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.
|||N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)|
|||L.A. [L.A. Lyusternik] Liusternik, "Elements of functional analysis" , F. Ungar (1961) (Translated from Russian)|
|||F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)|
|[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=11796