Symmetric operator

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A symmetric or formally self-adjoint operator is an object from mathematics . Such a linear operator is particularly considered in functional analysis in the context of unrestricted operators . Because a bounded symmetric operator is a self-adjoint operator .

In many applications, operators are considered that are unlimited . Examples are the momentum and Hamilton operators in quantum mechanics as well as many linear differential operators . In the case of unbounded differential operators, which are defined for bounded domains, it depends on the choice of the boundary conditions whether a symmetric differential operator is also essentially self-adjoint or even self-adjoint.


Be a Hilbert dream . A linear operator is called symmetric if

applies to all . With which is domain of designated.

The definition did not require that a symmetric operator be densely defined . However, there is only an operator to be adjoint if it is densely defined. Therefore, the definition of the symmetric operator in the literature is not uniform on this point.


  • A linear operator is symmetric if and only if holds.
  • For bounded linear operators, the terms self-adjoint and symmetric coincide. Therefore symmetric, not self-adjoint operators are always unbounded . In addition, Hellinger-Toeplitz's theorem says that every symmetric operator that is defined on the entire Hilbert space is continuous and therefore self-adjoint.
  • Semi-bounded operators are also symmetric. If a semi-bounded operator satisfies one of the inequalities
then he is even self-adjoint.
  • In contrast to the self-adjoint operators, symmetric operators can also have non-real eigenvalues.


Be the functional space of absolutely continuous functions on which vanish on the edge - so for valid. Since the space of absolutely continuous functions over a compact is isomorphic to the corresponding Sobolev space , the previously defined space can be understood as a Sobolev space with zero boundary conditions. Now consider the differential operator

into the Hilbert space of square integrable functions . This is symmetrical with respect to the complex -scalar product. This can be shown by means of partial integration . However, it is not self-adjoint, since the operator to be adjoint has, by definition, the maximum domain of definition, so the following applies to the adjoint operator


Here the functions in the domain of no longer meet the zero boundary condition. Another choice of the boundary condition of can make it a self-adjoint operator.

Individual evidence

  1. a b Dirk Werner : functional analysis. 6th, corrected edition, Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , p. 342.
  2. ^ A b Walter Rudin : Functional Analysis. McGraw-Hill, New York 1991. ISBN 0070542368 , p. 349.
  3. ^ Kosaku Yosida: Functional Analysis . 6th ed.Springer-Verlag, Berlin Heidelberg New York 1980, ISBN 3-540-10210-8 , pp. 197 .
  4. Dirk Werner : Functional Analysis. 6th, corrected edition, Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , p. 350.
  5. Dirk Werner : Functional Analysis. 6th, corrected edition, Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , p. 353.