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 ${\ displaystyle \ langle Tx, x \ rangle \ geq C \ | x \ | ^ {2}}$
• 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.