- Operador cerrado
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En matemáticas y específicamente en análisis funcional, los operadores lineales cerrados son un importante tipo de operadores lineales en los espacios de Banach. Son los más generales de los operadores acotados y, por tanto, no es necesario que la función sea continua, pero conserva suficientes buenas propiedades que pueden definir el espectro y partiendo de algún supuesto, el cálculo funcional para tales operadores. Muchos operadores lineales importantes no son acotados ni cerrados, tales como la derivada y ¿sus clases de operadores diferenciales?
Let B denote a Banach space. A linear operator
is closed if for every sequence
in 
converging to 
such that 
as 
one has 
and Ax = y. Equivalently, A is closed if its graph is closed in the direct sum 
Given a linear operator A, not necessarily closed, if the closure of its graph in
happens to be the graph of some operator, that operator is called the closure of A, and we say that A is closable. Denote the closure of A by 
It follows easily that A is the restriction of 
to 
A core of a closable operator is a subset
of such that the closure of the restriction of A to is The following properties are easily checked:
- Any closed linear operator defined on the whole space B is bounded. This is the closed graph theorem;
- If A is closed then A − λI is closed where λ is a scalar and I is the identity function;
- If A is closed and injective, then its inverse A − 1 is also closed;
- An operator A admits a closure if and only if for every pair of sequences {xn} and {yn} in converging to x and y, respectively, such that both {Axn} and {Ayn} converge, it holds lim nAxn = lim nAyn if x = y.
As an example, consider the derivative operator
where the Banach space B is the space C[a, b] of all continuous functions on an interval [a, b]. If one takes its domain
to be the largest set possible, that is, ![\mathcal{D}(A)=C^{1}[a, b],](3/e538a2a034843da0162268a9bab1b60a.png)
then A is a closed operator, which is not bounded.If one takes
to be instead the set of all infinitely differentiable functions, A will no longer be closed, but it will be closable, with the closure being its maximal extension defined on C1[a,b].See also densely defined operator and unbounded operator.
Categoría:- Operator theory
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