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A homomorphism of a certain function algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420301.png" /> (cf. [[Algebra of functions|Algebra of functions]]) into the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420302.png" /> of continuous linear operators on a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420303.png" />. A functional calculus is one of the basic tools of general spectral analysis and the theory of Banach algebras and it enables one to use function-analytic methods in these disciplines. Usually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420304.png" /> is a topological (in particular, normed) function algebra on a certain subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420305.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420306.png" /> containing the polynomials in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420307.png" /> (often as a dense subset), so that a functional calculus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420308.png" /> is a natural extension of the polynomial calculus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f0420309.png" /> in the commuting operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203011.png" />; in this case one says that the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203012.png" /> admits an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203014.png" />-calculus and one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203015.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203016.png" />-calculus for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203017.png" /> is a kind of spectral theorem, since the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203021.png" /> is the duality between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203023.png" />, determines a weak operator-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203024.png" />-distribution which commutes with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203025.png" />.
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The classical functional calculus of von Neumann–Murray–Dunford (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203027.png" /> is a reflexive space) leads to the operator (projection) [[Spectral measure|spectral measure]]
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203028.png" /></td> </tr></table>
+
A homomorphism of a certain function algebra  $  A $(
 +
cf. [[Algebra of functions|Algebra of functions]]) into the algebra  $  L ( X) $
 +
of continuous linear operators on a topological vector space  $  X $.
 +
A functional calculus is one of the basic tools of general spectral analysis and the theory of Banach algebras and it enables one to use function-analytic methods in these disciplines. Usually,  $  A $
 +
is a topological (in particular, normed) function algebra on a certain subset  $  K $
 +
of the space  $  \mathbf C  ^ {n} $
 +
containing the polynomials in the variables  $  z  ^ {1} \dots z  ^ {n} $(
 +
often as a dense subset), so that a functional calculus  $  \phi :  A \rightarrow L ( X) $
 +
is a natural extension of the polynomial calculus  $  p ( z  ^ {1} \dots z  ^ {n} ) \rightarrow p ( T _ {1} \dots T _ {n} ) $
 +
in the commuting operators  $  T _ {i} = \phi ( z  ^ {i} ) $,
 +
$  1 \leq  i \leq  n $;  
 +
in this case one says that the collection  $  T = ( T _ {1} \dots T _ {n} ) $
 +
admits an  $  A $-
 +
calculus and one writes  $  \phi ( T) = f ( T) = f ( T _ {1} \dots T _ {n} ) $.  
 +
An  $  A $-
 +
calculus for  $  T $
 +
is a kind of spectral theorem, since the correspondence  $  a \rightarrow \langle  \phi ( a) x, x  ^  \prime  \rangle $,
 +
where  $  x \in X $,
 +
$  x  ^  \prime  \in X  ^ {*} $
 +
and  $  \langle  , \rangle $
 +
is the duality between  $  X $
 +
and  $  X  ^ {*} $,
 +
determines a weak operator-valued  $  A $-
 +
distribution which commutes with  $  T $.
  
The functional calculus of Riesz–Dunford (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203030.png" />, that is, all functions holomorphic on the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203031.png" /> of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203032.png" />) leads to the formula
+
The classical functional calculus of von Neumann–Murray–Dunford ( $  A = C ( K) $,  
 +
$  X $
 +
is a reflexive space) leads to the operator (projection) [[Spectral measure|spectral measure]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203033.png" /></td> </tr></table>
+
$$
 +
\epsilon  = \epsilon _ {T} : \
 +
f ( T _ {1} \dots T _ {n} )  = \int\limits f d \epsilon .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203034.png" /> is the [[Resolvent|resolvent]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203036.png" /> is a contour enclosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203037.png" /> inside and on which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203038.png" /> is regular. Formulas of the latter type with several variables (operators) depend on the notation for a linear functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203039.png" /> and on the way the joint spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203040.png" /> of the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203041.png" /> is defined (the size of the functional calculus also depends on the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203042.png" />).
+
The functional calculus of Riesz–Dunford ( $  n = 1 $,
 +
$  A = \mathop{\rm Hol} ( \sigma ( T)) $,
 +
that is, all functions holomorphic on the spectrum $  \sigma ( T) $
 +
of the operator  $  T $)
 +
leads to the formula
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203043.png" /> is a [[Spectral operator|spectral operator]], if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203045.png" /> are its scalar and quasi-nilpotent parts, respectively, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203046.png" />, then the formula
+
$$
 +
f ( T)  = \
 +
{
 +
\frac{1}{2 \pi i }
 +
}
 +
\int\limits _  \gamma
 +
f ( \lambda ) R ( \lambda , T)  d \lambda ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203047.png" /></td> </tr></table>
+
where  $  R ( \lambda , T) = ( \lambda I - T)  ^ {-} 1 $
 +
is the [[Resolvent|resolvent]] of  $  T $
 +
and  $  \gamma $
 +
is a contour enclosing  $  \sigma ( T) $
 +
inside and on which the function  $  f $
 +
is regular. Formulas of the latter type with several variables (operators) depend on the notation for a linear functional on  $  \mathop{\rm Hol} ( \sigma ( T)) $
 +
and on the way the joint spectrum  $  \sigma ( T) $
 +
of the collection  $  T = ( T _ {1} \dots T _ {n} ) $
 +
is defined (the size of the functional calculus also depends on the definition of  $  \sigma ( T) $).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203048.png" /> is a [[Resolution of the identity|resolution of the identity]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203049.png" />, enables one to extend the Riesz–Dunford functional calculus for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203050.png" /> to a wider class of functions. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203052.png" /> admits a functional calculus on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203054.png" />-times continuously-differentiable functions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203055.png" /> is an operator of scalar type, then one can substitute bounded Borel functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203056.png" /> in this formula. In particular, the normal operators on a Hilbert space admit such a functional calculus. The converse is true: If an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203057.png" /> admits such a functional calculus (for operators in reflexive spaces it is sufficient to assume the existence of a functional calculus on the class of continuous functions), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203058.png" /> is a spectral operator of scalar type (in a Hilbert space this is a linear operator that is similar to a normal operator).
+
If  $  T $
 +
is a [[Spectral operator|spectral operator]], if $  S $
 +
and  $  N $
 +
are its scalar and quasi-nilpotent parts, respectively, and if  $  f \in \mathop{\rm Hol} ( \sigma ( T)) $,  
 +
then the formula
  
In [[#References|[5]]] the non-analytic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203059.png" />-calculus was constructed for operators with a resolvent of sufficiently slow growth near the spectrum; this was based on the Carleman classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203060.png" /> (cf. [[Quasi-analytic class|Quasi-analytic class]]) and used the formula
+
$$
 +
f ( T)  = \
 +
\sum _ {n \geq  0 }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203061.png" /></td> </tr></table>
+
\frac{N  ^ {n} }{n! }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203062.png" /> is the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203064.png" />-extension of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203065.png" /> across the boundary of the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203066.png" />, that is, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203067.png" />-function with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203068.png" /> for which
+
\int\limits _ {\sigma ( T) }
 +
f ^ { ( n) }  d \epsilon ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203069.png" /></td> </tr></table>
+
where  $  \epsilon $
 +
is a [[Resolution of the identity|resolution of the identity]] for  $  T $,
 +
enables one to extend the Riesz–Dunford functional calculus for  $  T $
 +
to a wider class of functions. In particular, if  $  N ^ {m + 1 } = 0 $,
 +
then  $  T $
 +
admits a functional calculus on the class  $  C  ^ {m} ( \sigma ( T)) $
 +
of  $  m $-
 +
times continuously-differentiable functions. If  $  T $
 +
is an operator of scalar type, then one can substitute bounded Borel functions on  $  \sigma ( T) $
 +
in this formula. In particular, the normal operators on a Hilbert space admit such a functional calculus. The converse is true: If an operator  $  T $
 +
admits such a functional calculus (for operators in reflexive spaces it is sufficient to assume the existence of a functional calculus on the class of continuous functions), then  $  T $
 +
is a spectral operator of scalar type (in a Hilbert space this is a linear operator that is similar to a normal operator).
 +
 
 +
In [[#References|[5]]] the non-analytic  $  C \{ M _ {k} \} $-
 +
calculus was constructed for operators with a resolvent of sufficiently slow growth near the spectrum; this was based on the Carleman classes  $  C ( \{ M _ {k} \} , \sigma ( T)) $(
 +
cf. [[Quasi-analytic class|Quasi-analytic class]]) and used the formula
 +
 
 +
$$
 +
f ( T) = - {
 +
\frac{1} \pi
 +
}
 +
{\int\limits \int\limits } _ {\mathbf C }
 +
 
 +
\frac{\partial  \widetilde{f}  }{\partial  \overline{z}\; }
 +
 
 +
( \lambda ) R ( \lambda , T)  d \lambda  \overline{ {d \lambda }}\; ,
 +
$$
 +
 
 +
where  $  \widetilde{f}  $
 +
is the so-called  $  \overline \partial \; $-
 +
extension of the function  $  f $
 +
across the boundary of the spectrum  $  \sigma ( T) $,
 +
that is, a  $  C  ^ {1} $-
 +
function with compact support in  $  \mathbf C $
 +
for which
 +
 
 +
$$
 +
\left . f = \widetilde{f}  \right | _ {\sigma ( T) }  ,\ \
 +
\left |
 +
 
 +
\frac{\partial  \widetilde{f}  }{\partial  \overline{z}\; }
 +
 
 +
( \lambda ) \right |  \leq  \
 +
\textrm{ const } \cdot
 +
h _ {\{ M _ {k}  \} } ( c  \mathop{\rm dist}  ( \lambda , K)).
 +
$$
  
 
Here
 
Here
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203070.png" /></td> </tr></table>
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203071.png" /></td> </tr></table>
+
\frac{\partial  \widetilde{f}  }{\partial  \overline{z}\; }
 +
  = \
 +
{
 +
\frac{1}{2}
 +
}
 +
\left (
  
and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203072.png" /> satisfies
+
\frac{\partial  \widetilde{f}  }{\partial  x }
 +
+ i
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203073.png" /></td> </tr></table>
+
\frac{\partial  \widetilde{f}  }{\partial  y }
  
On the other hand, bounds on the operator polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203074.png" /> lead to more extensive calculi (than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203075.png" />). For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203076.png" /> is a Hilbert space, then the von Neumann–Heinz inequality
+
\right ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203077.png" /></td> </tr></table>
+
$$
 +
h _ {\{ M _ {k}  \} } ( r)  = \inf _ { n }  r ^ {n - 1 }
 +
\frac{M _ {n} }{n ! }
 +
,
 +
$$
  
leads to the Szökefalvi-Nagy–Foias functional calculus (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203078.png" /> is the algebra of all holomorphic and bounded functions in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203080.png" /> is a contraction without unitary parts), which has many applications in the theory of functional models for contraction operators. The analogue of the von Neumann–Heinz inequality for symmetric function spaces provides a functional calculus in terms of multipliers (of corresponding convolution spaces [[#References|[8]]]).
+
and the operator  $  T $
 +
satisfies
  
Applications. The type of a functional calculus admitting an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203081.png" /> is invariant under a linear similarity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203082.png" /> and can be used successfully to classify operators. In particular, there is an extensive theory of the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203084.png" />-scalar operators, which can be applied to many classes of operators and is not confined to classical spectral theory. For a successful use of a functional calculus it is expedient to have the so-called spectral-mapping theorems:
+
$$
 +
\| R ( \lambda , T) \|  \leq  \
 +
\left (
 +
\frac{h _ {\{ M _ {k}  \} } (  \mathop{\rm dist}  ( \lambda , K) ) }{|  \mathop{\rm log}  \mathop{\rm dist}  ( \lambda , K) | }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203085.png" /></td> </tr></table>
+
\right ) .
 +
$$
 +
 
 +
On the other hand, bounds on the operator polynomials  $  p ( T) $
 +
lead to more extensive calculi (than  $  \mathop{\rm Hol} ( \sigma ( T)) $).
 +
For example, if  $  X $
 +
is a Hilbert space, then the von Neumann–Heinz inequality
 +
 
 +
$$
 +
\| p ( T) \|  \leq  \
 +
\max \{ {| p ( \xi ) | } : {| \xi | \leq  \| T \| } \}
 +
$$
 +
 
 +
leads to the Szökefalvi-Nagy–Foias functional calculus ( $  A $
 +
is the algebra of all holomorphic and bounded functions in the disc  $  \{ {\xi \in \mathbf C } : {| \xi | < 1 } \} $,
 +
$  T $
 +
is a contraction without unitary parts), which has many applications in the theory of functional models for contraction operators. The analogue of the von Neumann–Heinz inequality for symmetric function spaces provides a functional calculus in terms of multipliers (of corresponding convolution spaces [[#References|[8]]]).
 +
 
 +
Applications. The type of a functional calculus admitting an operator  $  T $
 +
is invariant under a linear similarity  $  T \rightarrow V  ^ {-} 1 TV $
 +
and can be used successfully to classify operators. In particular, there is an extensive theory of the so-called  $  A $-
 +
scalar operators, which can be applied to many classes of operators and is not confined to classical spectral theory. For a successful use of a functional calculus it is expedient to have the so-called spectral-mapping theorems:
 +
 
 +
$$
 +
\sigma ( f ( T))  = \
 +
f ( \sigma ( T)),\ \
 +
f \in A.
 +
$$
  
 
Such theorems have been proved for all the functional calculi listed above (after giving a suitable meaning to the right-hand side of the formula).
 
Such theorems have been proved for all the functional calculi listed above (after giving a suitable meaning to the right-hand side of the formula).
  
If the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203086.png" /> contains a fine partition of unity (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203087.png" />), then one can construct a local spectral analysis from an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203088.png" />-functional calculus and, in particular, one can prove the existence of non-trivial invariant subspaces of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203089.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203090.png" /> contains more than one point); an example is an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203091.png" /> (in a Banach space) with a spectrum that lies on a smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203093.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042030/f04203094.png" />. A corollary of the local analysis is Shilov's theorem on idempotents [[#References|[2]]].
+
If the algebra $  A $
 +
contains a fine partition of unity (for example, if $  A = C  ^  \infty  $),  
 +
then one can construct a local spectral analysis from an $  A $-
 +
functional calculus and, in particular, one can prove the existence of non-trivial invariant subspaces of the operator $  T $(
 +
if $  \sigma ( T) $
 +
contains more than one point); an example is an operator $  T $(
 +
in a Banach space) with a spectrum that lies on a smooth curve $  \gamma $
 +
and  $  \int _ {0}  ^  \infty  \mathop{\rm log}  ^ {+}  \mathop{\rm log}  ^ {+}  \delta ( r)  dr < \infty $,  
 +
where $  \delta ( r) = \max \{ {\| R ( \lambda , T) \| } : { \mathop{\rm dist} ( \lambda , \gamma ) \geq  r } \} $.  
 +
A corollary of the local analysis is Shilov's theorem on idempotents [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators" , '''1–3''' , Interscience  (1958–1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Spectral theories" , Addison-Wesley  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Waelbroeck,  "Etude spectrale des algèbres complètes"  ''Acad. Roy. Belgique Cl. Sci.'' , '''31''' :  7  (1960)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.L. Taylor,  "The analytic-functional calculus for several commuting operators"  ''Acta Math.'' , '''125''' :  1–2  (1970)  pp. 1–38</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.M. Dyn'kin,  "An operator calculus based on the Cauchy–Green formula"  ''Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst.'' , '''30'''  (1972)  pp. 33–39  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. von Neumann,  "Eine Spektraltheorie für allgemeine Operatoren eines unitären Räumes"  ''Math. Nachr.'' , '''4'''  (1950–1951)  pp. 258–281</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B. Szökefalvi-Nagy,  Ch. Foiaş,  "Harmonic analysis of operators on Hilbert space" , North-Holland  (1970)  (Translated from French)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.V. Peller,  "Estimates of operator polynomials in symmetric spaces. Functional calculus for absolute contraction operators"  ''Math. Notes'' , '''25'''  (1979)  pp. 464–471  ''Mat. Zametki'' , '''25''' :  6  (1979)  pp. 899–912</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  I. Colojoară,  C. Foiaş,  "Theory of generalized spectral operators" , Gordon &amp; Breach  (1968)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  Yu.I. Lyubich,  V.I. Matsaev,  "Operators with separable spectrum"  ''Mat. Sb.'' , '''56''' :  2  (1962)  pp. 433–468  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  J. Mikusiński,  "Operational calculus" , Pergamon  (1959)  (Translated from Polish)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  V.P. Maslov,  "Operational methods" , MIR  (1976)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators" , '''1–3''' , Interscience  (1958–1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Spectral theories" , Addison-Wesley  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Waelbroeck,  "Etude spectrale des algèbres complètes"  ''Acad. Roy. Belgique Cl. Sci.'' , '''31''' :  7  (1960)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.L. Taylor,  "The analytic-functional calculus for several commuting operators"  ''Acta Math.'' , '''125''' :  1–2  (1970)  pp. 1–38</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.M. Dyn'kin,  "An operator calculus based on the Cauchy–Green formula"  ''Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst.'' , '''30'''  (1972)  pp. 33–39  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. von Neumann,  "Eine Spektraltheorie für allgemeine Operatoren eines unitären Räumes"  ''Math. Nachr.'' , '''4'''  (1950–1951)  pp. 258–281</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B. Szökefalvi-Nagy,  Ch. Foiaş,  "Harmonic analysis of operators on Hilbert space" , North-Holland  (1970)  (Translated from French)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.V. Peller,  "Estimates of operator polynomials in symmetric spaces. Functional calculus for absolute contraction operators"  ''Math. Notes'' , '''25'''  (1979)  pp. 464–471  ''Mat. Zametki'' , '''25''' :  6  (1979)  pp. 899–912</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  I. Colojoară,  C. Foiaş,  "Theory of generalized spectral operators" , Gordon &amp; Breach  (1968)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  Yu.I. Lyubich,  V.I. Matsaev,  "Operators with separable spectrum"  ''Mat. Sb.'' , '''56''' :  2  (1962)  pp. 433–468  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  J. Mikusiński,  "Operational calculus" , Pergamon  (1959)  (Translated from Polish)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  V.P. Maslov,  "Operational methods" , MIR  (1976)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:40, 5 June 2020


A homomorphism of a certain function algebra $ A $( cf. Algebra of functions) into the algebra $ L ( X) $ of continuous linear operators on a topological vector space $ X $. A functional calculus is one of the basic tools of general spectral analysis and the theory of Banach algebras and it enables one to use function-analytic methods in these disciplines. Usually, $ A $ is a topological (in particular, normed) function algebra on a certain subset $ K $ of the space $ \mathbf C ^ {n} $ containing the polynomials in the variables $ z ^ {1} \dots z ^ {n} $( often as a dense subset), so that a functional calculus $ \phi : A \rightarrow L ( X) $ is a natural extension of the polynomial calculus $ p ( z ^ {1} \dots z ^ {n} ) \rightarrow p ( T _ {1} \dots T _ {n} ) $ in the commuting operators $ T _ {i} = \phi ( z ^ {i} ) $, $ 1 \leq i \leq n $; in this case one says that the collection $ T = ( T _ {1} \dots T _ {n} ) $ admits an $ A $- calculus and one writes $ \phi ( T) = f ( T) = f ( T _ {1} \dots T _ {n} ) $. An $ A $- calculus for $ T $ is a kind of spectral theorem, since the correspondence $ a \rightarrow \langle \phi ( a) x, x ^ \prime \rangle $, where $ x \in X $, $ x ^ \prime \in X ^ {*} $ and $ \langle , \rangle $ is the duality between $ X $ and $ X ^ {*} $, determines a weak operator-valued $ A $- distribution which commutes with $ T $.

The classical functional calculus of von Neumann–Murray–Dunford ( $ A = C ( K) $, $ X $ is a reflexive space) leads to the operator (projection) spectral measure

$$ \epsilon = \epsilon _ {T} : \ f ( T _ {1} \dots T _ {n} ) = \int\limits f d \epsilon . $$

The functional calculus of Riesz–Dunford ( $ n = 1 $, $ A = \mathop{\rm Hol} ( \sigma ( T)) $, that is, all functions holomorphic on the spectrum $ \sigma ( T) $ of the operator $ T $) leads to the formula

$$ f ( T) = \ { \frac{1}{2 \pi i } } \int\limits _ \gamma f ( \lambda ) R ( \lambda , T) d \lambda , $$

where $ R ( \lambda , T) = ( \lambda I - T) ^ {-} 1 $ is the resolvent of $ T $ and $ \gamma $ is a contour enclosing $ \sigma ( T) $ inside and on which the function $ f $ is regular. Formulas of the latter type with several variables (operators) depend on the notation for a linear functional on $ \mathop{\rm Hol} ( \sigma ( T)) $ and on the way the joint spectrum $ \sigma ( T) $ of the collection $ T = ( T _ {1} \dots T _ {n} ) $ is defined (the size of the functional calculus also depends on the definition of $ \sigma ( T) $).

If $ T $ is a spectral operator, if $ S $ and $ N $ are its scalar and quasi-nilpotent parts, respectively, and if $ f \in \mathop{\rm Hol} ( \sigma ( T)) $, then the formula

$$ f ( T) = \ \sum _ {n \geq 0 } \frac{N ^ {n} }{n! } \int\limits _ {\sigma ( T) } f ^ { ( n) } d \epsilon , $$

where $ \epsilon $ is a resolution of the identity for $ T $, enables one to extend the Riesz–Dunford functional calculus for $ T $ to a wider class of functions. In particular, if $ N ^ {m + 1 } = 0 $, then $ T $ admits a functional calculus on the class $ C ^ {m} ( \sigma ( T)) $ of $ m $- times continuously-differentiable functions. If $ T $ is an operator of scalar type, then one can substitute bounded Borel functions on $ \sigma ( T) $ in this formula. In particular, the normal operators on a Hilbert space admit such a functional calculus. The converse is true: If an operator $ T $ admits such a functional calculus (for operators in reflexive spaces it is sufficient to assume the existence of a functional calculus on the class of continuous functions), then $ T $ is a spectral operator of scalar type (in a Hilbert space this is a linear operator that is similar to a normal operator).

In [5] the non-analytic $ C \{ M _ {k} \} $- calculus was constructed for operators with a resolvent of sufficiently slow growth near the spectrum; this was based on the Carleman classes $ C ( \{ M _ {k} \} , \sigma ( T)) $( cf. Quasi-analytic class) and used the formula

$$ f ( T) = - { \frac{1} \pi } {\int\limits \int\limits } _ {\mathbf C } \frac{\partial \widetilde{f} }{\partial \overline{z}\; } ( \lambda ) R ( \lambda , T) d \lambda \overline{ {d \lambda }}\; , $$

where $ \widetilde{f} $ is the so-called $ \overline \partial \; $- extension of the function $ f $ across the boundary of the spectrum $ \sigma ( T) $, that is, a $ C ^ {1} $- function with compact support in $ \mathbf C $ for which

$$ \left . f = \widetilde{f} \right | _ {\sigma ( T) } ,\ \ \left | \frac{\partial \widetilde{f} }{\partial \overline{z}\; } ( \lambda ) \right | \leq \ \textrm{ const } \cdot h _ {\{ M _ {k} \} } ( c \mathop{\rm dist} ( \lambda , K)). $$

Here

$$ \frac{\partial \widetilde{f} }{\partial \overline{z}\; } = \ { \frac{1}{2} } \left ( \frac{\partial \widetilde{f} }{\partial x } + i \frac{\partial \widetilde{f} }{\partial y } \right ) , $$

$$ h _ {\{ M _ {k} \} } ( r) = \inf _ { n } r ^ {n - 1 } \frac{M _ {n} }{n ! } , $$

and the operator $ T $ satisfies

$$ \| R ( \lambda , T) \| \leq \ \left ( \frac{h _ {\{ M _ {k} \} } ( \mathop{\rm dist} ( \lambda , K) ) }{| \mathop{\rm log} \mathop{\rm dist} ( \lambda , K) | } \right ) . $$

On the other hand, bounds on the operator polynomials $ p ( T) $ lead to more extensive calculi (than $ \mathop{\rm Hol} ( \sigma ( T)) $). For example, if $ X $ is a Hilbert space, then the von Neumann–Heinz inequality

$$ \| p ( T) \| \leq \ \max \{ {| p ( \xi ) | } : {| \xi | \leq \| T \| } \} $$

leads to the Szökefalvi-Nagy–Foias functional calculus ( $ A $ is the algebra of all holomorphic and bounded functions in the disc $ \{ {\xi \in \mathbf C } : {| \xi | < 1 } \} $, $ T $ is a contraction without unitary parts), which has many applications in the theory of functional models for contraction operators. The analogue of the von Neumann–Heinz inequality for symmetric function spaces provides a functional calculus in terms of multipliers (of corresponding convolution spaces [8]).

Applications. The type of a functional calculus admitting an operator $ T $ is invariant under a linear similarity $ T \rightarrow V ^ {-} 1 TV $ and can be used successfully to classify operators. In particular, there is an extensive theory of the so-called $ A $- scalar operators, which can be applied to many classes of operators and is not confined to classical spectral theory. For a successful use of a functional calculus it is expedient to have the so-called spectral-mapping theorems:

$$ \sigma ( f ( T)) = \ f ( \sigma ( T)),\ \ f \in A. $$

Such theorems have been proved for all the functional calculi listed above (after giving a suitable meaning to the right-hand side of the formula).

If the algebra $ A $ contains a fine partition of unity (for example, if $ A = C ^ \infty $), then one can construct a local spectral analysis from an $ A $- functional calculus and, in particular, one can prove the existence of non-trivial invariant subspaces of the operator $ T $( if $ \sigma ( T) $ contains more than one point); an example is an operator $ T $( in a Banach space) with a spectrum that lies on a smooth curve $ \gamma $ and $ \int _ {0} ^ \infty \mathop{\rm log} ^ {+} \mathop{\rm log} ^ {+} \delta ( r) dr < \infty $, where $ \delta ( r) = \max \{ {\| R ( \lambda , T) \| } : { \mathop{\rm dist} ( \lambda , \gamma ) \geq r } \} $. A corollary of the local analysis is Shilov's theorem on idempotents [2].

References

[1] N. Dunford, J.T. Schwartz, "Linear operators" , 1–3 , Interscience (1958–1971)
[2] N. Bourbaki, "Elements of mathematics. Spectral theories" , Addison-Wesley (1977) (Translated from French)
[3] L. Waelbroeck, "Etude spectrale des algèbres complètes" Acad. Roy. Belgique Cl. Sci. , 31 : 7 (1960)
[4] J.L. Taylor, "The analytic-functional calculus for several commuting operators" Acta Math. , 125 : 1–2 (1970) pp. 1–38
[5] E.M. Dyn'kin, "An operator calculus based on the Cauchy–Green formula" Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst. , 30 (1972) pp. 33–39 (In Russian)
[6] J. von Neumann, "Eine Spektraltheorie für allgemeine Operatoren eines unitären Räumes" Math. Nachr. , 4 (1950–1951) pp. 258–281
[7] B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French)
[8] V.V. Peller, "Estimates of operator polynomials in symmetric spaces. Functional calculus for absolute contraction operators" Math. Notes , 25 (1979) pp. 464–471 Mat. Zametki , 25 : 6 (1979) pp. 899–912
[9] I. Colojoară, C. Foiaş, "Theory of generalized spectral operators" , Gordon & Breach (1968)
[10] Yu.I. Lyubich, V.I. Matsaev, "Operators with separable spectrum" Mat. Sb. , 56 : 2 (1962) pp. 433–468 (In Russian)
[11] J. Mikusiński, "Operational calculus" , Pergamon (1959) (Translated from Polish)
[12] V.P. Maslov, "Operational methods" , MIR (1976) (Translated from Russian)

Comments

For a systematic treatment of analytic functional calculi in several variables cf. [a1].

References

[a1] F.H. Vasilescu, "Analytic functional calculus and spectral decompositions" , Reidel & Ed. Academici (1982)
How to Cite This Entry:
Functional calculus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional_calculus&oldid=14302
This article was adapted from an original article by N.K. Nikol'skiiV.V. Peller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article