Symmetric operator

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.

definition

Be a Hilbert dream . A linear operator is called symmetric if ${\displaystyle H}$ ${\displaystyle T\colon D(T)\to H}$

${\displaystyle \langle Ty,x\rangle =\langle y,Tx\rangle }$

applies to all . With which is domain of designated. ${\displaystyle x,\,y\in D(T)}$ ${\displaystyle D(T)\subset H}$ ${\displaystyle T}$

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. ${\displaystyle T}$ ${\displaystyle T}$

properties

• A linear operator is symmetric if and only if holds. ${\displaystyle T}$ ${\displaystyle T\subseteq T^{*}}$

• 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 ${\displaystyle T}$

${\displaystyle \langle Tx,x\rangle \leq C\|x\|^{2}}$ or ${\displaystyle \langle Tx,x\rangle \geq C\|x\|^{2}}$

then he is even self-adjoint.

• In contrast to the self-adjoint operators, symmetric operators can also have non-real eigenvalues.

example

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 ${\displaystyle D}$ ${\displaystyle f}$ ${\displaystyle [0,1]}$ ${\displaystyle f(0)=f(1)=0}$ ${\displaystyle W^{1,1}}$ ${\displaystyle D}$ ${\displaystyle W_{0}^{1,1}([0,1])}$

${\displaystyle i{\frac {\mathrm {d} }{\mathrm {d} t}}\colon D\to L^{2}([0,1])}$

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 ${\displaystyle L^{2}([0,1])}$ ${\displaystyle L^{2}}$ ${\displaystyle i{\tfrac {\mathrm {d} }{\mathrm {d} t}}\colon D\to L^{2}([0,1])}$ ${\displaystyle i{\tfrac {\mathrm {d} }{\mathrm {d} t}}}$

${\displaystyle \left(i{\frac {\mathrm {d} }{\mathrm {d} t}}\right)^{*}\colon W^{1,1}([0,1])\to L^{2}([0,1])}$.

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. ${\displaystyle \left(i{\tfrac {\mathrm {d} }{\mathrm {d} t}}\right)^{*}}$ ${\displaystyle i{\tfrac {\mathrm {d} }{\mathrm {d} t}}}$

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.