Deficiency subspace

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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.


[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)


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).

How to Cite This Entry:
Deficiency subspace. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article