Closed linear operator | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-06

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Unbounded_operator


00:01:52 1 Short history
00:02:31 2 Definitions and basic properties
00:05:39 3 Example
00:13:57 4 Adjoint
00:20:30 5 Transpose
00:23:02 6 Closed linear operators
00:23:15 6.1 Basic properties
00:23:27 6.2 Example
00:27:07 7 Symmetric operators and self-adjoint operators
00:29:00 8 Extension-related
00:30:03 9 Importance of self-adjoint operators
00:30:07 10 See also
00:30:16 11 Notes
00:30:30 12 References
00:30:59 C∞([a, b]), then A will no longer be closed, but it will be closable, with the closure being its extension defined on C1([a, b]).
00:31:14 Symmetric operators and self-adjoint operators
00:33:25 x and Tz + iz
00:35:56 Extension-related
00:40:03 T∗∗
00:40:18 T∗.
00:40:22 Importance of self-adjoint operators
00:40:34 The class of self-adjoint operators is especially important in mathematical physics. Every self-adjoint operator is densely defined, closed and symmetric. The converse holds for bounded operators but fails in general. Self-adjointness is substantially more restricting than these three properties. The famous spectral theorem holds for self-adjoint operators. In combination with Stone's theorem on one-parameter unitary groups it shows that self-adjoint operators are precisely the infinitesimal generators of strongly continuous one-parameter unitary groups, see Self-adjoint operator#Self-adjoint extensions in quantum mechanics. Such unitary groups are especially important for describing time evolution in classical and quantum mechanics.
00:41:40 See also



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SUMMARY
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In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
The term "unbounded operator" can be misleading, since

"unbounded" should sometimes be understood as "not necessarily bounded";
"operator" should be understood as "linear operator" (as in the case of "bounded operator");
the domain of the operator is a linear subspace, not necessarily the whole space;
this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense;
in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain.
The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above. The given space is assumed to be a Hilbert space. Some generalizations to Banach spaces and more general topological vector spaces are possible.