# Deficiency subspace

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defect subspace, defective subspace, of an operator

The orthogonal complement $D _ \lambda$ of the range of values $T _ \lambda = \{ {y = ( A - \lambda I ) x } : {x \in D _ {A} } \}$ of the operator $A _ \lambda = A - \lambda I$, where $A$ is a linear operator defined on a linear manifold $D _ {A}$ of a Hilbert space $H$, while $\lambda$ is a regular value (regular point) of $A$. Here, a regular value of an operator $A$ is understood to be a value of the parameter $\lambda$ for which the equation $( A - \lambda I ) x = y$ has a unique solution for any $y$ while the operator $( A - \lambda I ) ^ {-1}$ is bounded, i.e. the resolvent of $A$ is bounded. As $\lambda$ changes, the deficiency subspace $D _ \lambda$ changes as well, but its dimension remains the same for all $\lambda$ belonging to a connected component of the open set of all regular values of $A$.

If $A$ is a symmetric operator with a dense domain of definition $D _ {A}$, its connected components of regular values will be the upper and the lower half-plane. In this case $D _ \lambda = \{ {x \in D _ {A ^ {*} } } : {A ^ {*} x = \overline \lambda \; x } \}$, while the deficiency numbers $n _ {+} = \mathop{\rm dim} D _ {i}$ and $n _ {-} = \mathop{\rm dim} D _ {-i}$, where $A ^ {*}$ is the adjoint operator, are called the (positive and negative) deficiency indices of the operator $A$. In addition,

$$D _ {A ^ {*} } = D _ {A} \oplus D _ {i} \oplus D _ {-i} ,$$

i.e. $D _ {A ^ {*} }$ is the direct sum of $D _ {A}$, $D _ {i}$ and $D _ {-i}$. Thus, if $n _ {+} = n _ {-} = 0$, the operator $A$ 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)

The definition of a regular value of an operator as given above is not quite correct and should read as follows. The value $\lambda$ is a regular value of $A$ if there exists a positive number $k = k ( \lambda ) > 0$ such that $\| ( A - \lambda I ) x \| \geq k \| x \|$ for all $x \in D _ {A}$. In that case the kernel of $A - \lambda I$ consists of the zero vector only and the image of $A - \lambda I$ is closed (but not necessarily equal to the whole space).