View source for Closed operator

An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c0225701.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c0225702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c0225703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c0225704.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c0225705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c0225706.png" />. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c0225707.png" /> are Banach spaces over the same field of scalars and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c0225708.png" /> is the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c0225709.png" />.) The notion of a closed operator may be extended to operators defined on separable linear topological spaces, except that instead of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257010.png" /> one must consider arbitrary directions (nets) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257012.png" /> is the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257014.png" /> is closed if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257015.png" /> is a closed subset of the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257016.png" />. This property is often adopted as the definition of a closed operator.

The notion of a closed operator is a generalization of the notion of an operator defined and continuous on a closed subset of a Banach space. An example of a closed but not continuous operator is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257017.png" />, defined on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257018.png" /> of continuously-differentiable functions in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257019.png" />. Many operators of mathematical physics are closed but not continuous.

An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257020.png" /> has a closure (i.e. is closeable) if it admits a closed extension. An operator has a closure if and only if it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257021.png" />,

<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/c/c022/c022570/c02257022.png" /></td> </tr></table>

that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257023.png" />. The smallest closed extension of an operator is called its closure. A symmetric operator on a Hilbert space with dense domain of definition always admits a closure.

A bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257024.png" /> is closed. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257025.png" /> is defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257026.png" /> and closed, then it is bounded. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257027.png" /> is closed and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257028.png" /> exists, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257029.png" /> is also closed. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257030.png" /> is closed if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257031.png" /> is closed, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257032.png" /> is closed if the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257033.png" /> exists and is bounded for at least one value of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257034.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257035.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257036.png" /> and, consequently, the adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257038.png" />, is uniquely defined, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257039.png" /> is a closed operator. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257040.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257042.png" /> are reflexive, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257043.png" /> is a closeable operator and its closure is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257044.png" />.

A closed operator can be made bounded by introducing a new norm on its domain of definition. Let

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Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257046.png" /> with this new norm is a Banach space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257047.png" />, as an operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257048.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022570/c02257049.png" />, is bounded.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1980)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T. Kato,   "Perturbation theory for linear operators" , Springer  (1980)</TD></TR></table>



====Comments====
The result that a closed linear operator mapping (all of) a Banach space into a Banach space is continuous is known as the closed-graph theorem.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Goldberg,   "Unbounded linear operators" , McGraw-Hill  (1966)</TD></TR></table>