en>RomanSpa at 18:23, 21 February 2014

2014-02-21T18:23:51Z

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	~~{{Unreferenced\|date=December 2009}}~~		Hello and welcome. My title is Irwin and I completely dig that name. What I love doing is performing ceramics but I haven't made a dime with it. Managing individuals has been his day occupation for a while. Puerto Rico is where he's usually been living but she needs to move because of her family members.<br><br>Feel free to visit my homepage; [http://www.hotporn123.com/blog/150819 hotporn123.com]
	~~In [[mathematics]], in the area of [[functional analysis]]~~ and [[operator theory]], the '''Volterra operator''', named after [[Vito Volterra]], represents the operation of [[indefinite integration]], viewed as a [[bounded linear operator]] on the space ''L''<sup>2</sup>(0,1) of complex-valued [[square integrable function]]s on the interval (0,1). It is ~~the operator corresponding to the [[Volterra integral equation]]s~~.

	~~==Definition==~~
	~~The Volterra operator,~~ '~~'V'', may be defined~~ for a ~~function ''f'' ∈ ''L'~~'~~<sup>2</sup>(0,1) and a value ''t'' ∈ (0,1), as~~

	~~:<math>V(f)(t) = \int_0^t{f(~~s~~)\, ds}~~.<~~/math~~>

	~~==Properties==~~
	*''V'' is a bounded linear operator between Hilbert spaces, with [[Hermitian adjoint]]

	::<~~math~~>~~V^*(f)(t) = \int_t^1{f(s)\, ds}.</math>~~
	*''V'' is a [[Hilbert-Schmidt operator]], hence in particular is [[~~compact operator\|compact]].<ref name='stackexchange_indefinite-integrators'>{{cite web\|title=Spectrum of Indefinite Integral Operators (From stackexchange.com)\|url=~~http://~~math~~.~~stackexchange~~.com/~~questions~~/~~151425/spectrum-of-indefinite-integral-operators}}</ref>~~
	*''V'' has no [[eigenvalue]]s and therefore, by the [[spectral theory of compact operators]], its [[spectrum (functional analysis)\|spectrum]] σ(''V'') = {0}.~~<ref name='stackexchange_indefinite-integrators' />~~
	*''V'' is a [[quasinilpotent operator]] (that is, the [[spectral radius]], ''ρ''(''V''), is zero), but it is not [[nilpotent]].
	*The [[operator norm]] of ''V'' is exactly \|\|''V''\|\| = <sup>2</sup>⁄<sub>π</sub>.<ref name='stackexchange_indefinite-integrators' />

	~~==References==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Volterra Operator}}~~
	~~[[Category:Operator theory]~~]

en>Myasuda: added missing diacritics

2012-04-30T00:21:27Z

added missing diacritics

New page

{{Unreferenced|date=December 2009}}
In [[mathematics]], in the area of [[functional analysis]] and [[operator theory]], the '''Volterra operator''', named after [[Vito Volterra]], represents the operation of [[indefinite integration]], viewed as a [[bounded linear operator]] on the space ''L''2(0,1) of complex-valued [[square integrable function]]s on the interval (0,1). It is the operator corresponding to the [[Volterra integral equation]]s.

==Definition==
The Volterra operator, ''V'', may be defined for a function ''f'' ∈ ''L''2(0,1) and a value ''t'' ∈ (0,1), as

:<math>V(f)(t) = \int_0^t{f(s)\, ds}.</math>

==Properties==
*''V'' is a bounded linear operator between Hilbert spaces, with [[Hermitian adjoint]]

::<math>V^*(f)(t) = \int_t^1{f(s)\, ds}.</math>
*''V'' is a [[Hilbert-Schmidt operator]], hence in particular is [[compact operator|compact]].<ref name='stackexchange_indefinite-integrators'>{{cite web|title=Spectrum of Indefinite Integral Operators (From stackexchange.com)|url=http://math.stackexchange.com/questions/151425/spectrum-of-indefinite-integral-operators}}</ref>
*''V'' has no [[eigenvalue]]s and therefore, by the [[spectral theory of compact operators]], its [[spectrum (functional analysis)|spectrum]] σ(''V'') = {0}.<ref name='stackexchange_indefinite-integrators' />
*''V'' is a [[quasinilpotent operator]] (that is, the [[spectral radius]], ''ρ''(''V''), is zero), but it is not [[nilpotent]].
*The [[operator norm]] of ''V'' is exactly ||''V''|| = 2⁄π.<ref name='stackexchange_indefinite-integrators' />

==References==
{{Reflist}}

{{DEFAULTSORT:Volterra Operator}}
[[Category:Operator theory]]

Foucault pendulum vector diagrams - Revision history

en>RomanSpa at 18:23, 21 February 2014

en>Myasuda: added missing diacritics