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Difference between revisions of "Deficiency subspace"

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for which the equation  $  ( A - \lambda I ) x = y $
 
for which the equation  $  ( A - \lambda I ) x = y $
 
has a unique solution for any  $  y $
 
has a unique solution for any  $  y $
while the operator  $  ( A - \lambda I )  ^ {-} 1 $
+
while the operator  $  ( A - \lambda I )  ^ {-1} $
 
is bounded, i.e. the [[Resolvent|resolvent]] of  $  A $
 
is bounded, i.e. the [[Resolvent|resolvent]] of  $  A $
 
is bounded. As  $  \lambda $
 
is bounded. As  $  \lambda $
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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 } \} $,  
 
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} $
 
while the deficiency numbers  $  n _ {+} =  \mathop{\rm dim}  D _ {i} $
and  $  n _ {-} =  \mathop{\rm dim}  D _ {-} i $,  
+
and  $  n _ {-} =  \mathop{\rm dim}  D _ {-i} $,  
 
where  $  A  ^ {*} $
 
where  $  A  ^ {*} $
 
is the adjoint operator, are called the (positive and negative) deficiency indices of the operator  $  A $.  
 
is the adjoint operator, are called the (positive and negative) deficiency indices of the operator  $  A $.  
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$$  
 
$$  
D _ {A  ^ {*}  }  =  D _ {A} \oplus D _ {i} \oplus D _ {-} i ,
+
D _ {A  ^ {*}  }  =  D _ {A} \oplus D _ {i} \oplus D _ {-i} ,
 
$$
 
$$
  
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is the direct sum of  $  D _ {A} $,  
 
is the direct sum of  $  D _ {A} $,  
 
$  D _ {i} $
 
$  D _ {i} $
and  $  D _ {-} i $.  
+
and  $  D _ {-i} $.  
 
Thus, if  $  n _ {+} = n _ {-} = 0 $,  
 
Thus, if  $  n _ {+} = n _ {-} = 0 $,  
 
the operator  $  A $
 
the operator  $  A $

Latest revision as of 17:06, 19 January 2024


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)

Comments

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).

How to Cite This Entry:
Deficiency subspace. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Deficiency_subspace&oldid=55209
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article