properties of the adjoint operator

Let A and B be linear operatorsMathworldPlanetmath in a Hilbert spaceMathworldPlanetmath, and let λ. Assuming all the operators involved are densely defined, the following properties hold:

  1. 1.

    If A-1 exists and is densely defined, then (A-1)*=(A*)-1;

  2. 2.


  3. 3.

    AB implies B*A*;

  4. 4.


  5. 5.


  6. 6.


  7. 7.

    A* is a closed operatorMathworldPlanetmath.

Remark. The notation AB for operators means that B is an of A, i.e. A is the restriction ( of B to a smaller domain.

Also, we have the following

Proposition 1

If A admits a closurePlanetmathPlanetmath ( A¯, then A* is densely defined and (A*)*=A¯.

Title properties of the adjoint operator
Canonical name PropertiesOfTheAdjointOperator
Date of creation 2013-03-22 13:48:14
Last modified on 2013-03-22 13:48:14
Owner Koro (127)
Last modified by Koro (127)
Numerical id 12
Author Koro (127)
Entry type Theorem
Classification msc 47A05