Deficiency subspace
defect subspace, defective subspace, of an operator
The orthogonal complement of the range of values
of the operator
, where
is a linear operator defined on a linear manifold
of a Hilbert space
, while
is a regular value (regular point) of
. Here, a regular value of an operator
is understood to be a value of the parameter
for which the equation
has a unique solution for any
while the operator
is bounded, i.e. the resolvent of
is bounded. As
changes, the deficiency subspace
changes as well, but its dimension remains the same for all
belonging to a connected component of the open set of all regular values of
.
If is a symmetric operator with a dense domain of definition
, its connected components of regular values will be the upper and the lower half-plane. In this case
, while the deficiency numbers
and
, where
is the adjoint operator, are called the (positive and negative) deficiency indices of the operator
. In addition,
![]() |
i.e. is the direct sum of
,
and
. Thus, if
, the operator
is self-adjoint; otherwise the deficiency subspace of a symmetric operator characterizes the extent of its deviation from a self-adjoint operator.
Deficiency subspaces play an important role in constructing the extensions of a symmetric operator to a maximal operator or to a self-adjoint (hyper-maximal) operator.
References
[1] | L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Hindushtan Publ. Comp. (1974) (Translated from Russian) |
[2] | N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in a Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian) |
[3] | N. Dunford, J.T. Schwartz, "Linear operators" , 1–2 , Interscience (1958–1963) |
[4] | F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French) |
Comments
The definition of a regular value of an operator as given above is not quite correct and should read as follows. The value is a regular value of
if there exists a positive number
such that
for all
. In that case the kernel of
consists of the zero vector only and the image of
is closed (but not necessarily equal to the whole space).
Deficiency subspace. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Deficiency_subspace&oldid=15718