In the model, the operator " T " is multiplication by " x " and a densely defined symmetric operator.
12.
The spectrum of any bounded symmetric operator is real; in particular all its eigenvalues are real, although a symmetric operator may have no eigenvalues.
13.
The spectrum of any bounded symmetric operator is real; in particular all its eigenvalues are real, although a symmetric operator may have no eigenvalues.
14.
A densely defined, symmetric operator " T " is essentially self-adjoint if and only if both operators, have dense range.
15.
According to the reciprocal relations, the damped oscillations in homogeneous closed systems near thermodynamic equilibria are impossible because the spectrum of symmetric operators is real.
16.
More specifically, finding self-adjoint extensions, with various requirements, of symmetric operators is equivalent to finding unitary extensions of suitable partial isometries.
17.
A densely defined closed symmetric operator " T " is self-adjoint if and only if " T " " is symmetric.
18.
As was pointed out in a previous example, a sufficient condition that an unbounded symmetric operator has eigenvectors which form a Hilbert space basis is that it has a compact inverse.
19.
If the set of eigenvalues for a symmetric operator is non empty, and the eigenvalues are nondegenerate, then it follows from the definition that eigenvectors corresponding to distinct eigenvalues are orthogonal.
20.
A general version of the spectral theorem which also applies to bounded symmetric operators ( see Reed and Simon, vol . 1, chapter VII, or other books cited ) is stated below.
