Deficiency subspace

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.

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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).