0.1 Definition 1
Let be a finite dimensional Hilbert space. A linear operator is said to be diagonalizable if the corresponding matrix (in a given basis) is a diagonalizable matrix (http://planetmath.org/Diagonalizable2).
Remark - This is a common definition in linear algebra.
0.2 Definition 2
The above definition is equivalent to: There exists an orthonormal basis of consisting of eigenvectors of .
Another equivalent definition is: There exists an orthonormal basis of and values such that
In linear algebra (http://planetmath.org/LinearAlgebra) such operators are also called unitarily diagonalizable.
0.3 Definition 3
Let be a Hilbert space. A bounded linear operator is said to be diagonalizable if there exists an orthonormal basis consisting of eigenvectors of .
An equivalent definition is: There exists an orthonormal basis of and values such that
0.4 Definition 4
Let be a Hilbert space. A linear operator is said to be diagonalizable if it is to a multiplication operator (http://planetmath.org/MultiplicationOperatorOnMathbbL22) in some -space (http://planetmath.org/L2SpacesAreHilbertSpaces), i.e. if there exists
where is the operator of multiplication (http://planetmath.org/MultiplicationOperator) by
If the above definition is equivalent to say that is unitarily diagonalizable (Definition 2). Indeed, we can think of as with the counting measure. In this case, multiplication operators correspond to diagonal matrices.
Diagonalizable operators (in this sense) are necessarily normal operators (since multiplication operators are so). The discussion about the converse result is the content of general versions of the spectral theorem.
|Date of creation||2013-03-22 17:33:47|
|Last modified on||2013-03-22 17:33:47|
|Last modified by||asteroid (17536)|