imported>Mateusz: added "Functional calculus"

2012-07-20T11:50:50Z

added "Functional calculus"

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	Conversely, for any Bernstein function $f$ there exists a semigroup of sub-probabilistic measures $\mu_t$ satisfying the above equality.		Conversely, for any Bernstein function $f$ there exists a semigroup of sub-probabilistic measures $\mu_t$ satisfying the above equality.

	~~One often writes $\tilde{L}~~ = ~~f(L)$, where~~ $-L$ and $-\tilde{L}$ ~~are~~ the generators of $T_t$ and $\tilde{T}_t$, respectively. This notation agrees with spectral-theoretic definition of $f(L)$ if the underlying Banach space is a Hilbert space.		==Functional calculus==
			Let $-L$ and $-\tilde{L}$ be the generators of $T_t$ and $\tilde{T}_t$, respectively. If $f$ is the Bernstein function described above, then it is customary to write $\tilde{L} = f(L)$.

			This notation agrees with spectral-theoretic definition of $f(L)$ if the underlying Banach space is a Hilbert space. Furthermore, if $f_1$, $f_2$ are Bernstein functions and $c > 0$, then
			\begin{align} c f_1(L) & = (c f_1)(L) , \\ f_1(L) + f_2(L) & = (f_1 + f_2)(L) , \\ f_1(f_2(L)) & = (f_1 \circ f_2)(L) , \end{align}
			and if $f_1 f_2$ is also a Bernstein function, then
			\[ f_1(L) f_2(L) = (f_1 f_2)(L) . \]

			Since $z^s$ is a Bernstein function for $s \in (0, 1)$, subordination gives a way to define the fractional power $L^s$ of an operator $L$ on a Banach space, provided that $-L$ generates a strongly continuous semigroup of contractions.

	==References==		==References==

imported>Mateusz: Created page with "If $T_t$ is a strongly continuous semigroup of contraction operators on a Banach space and $\mu_t$ is a semigroup of sub-probabilistic measures on $[0, \infty)$, then \[ \tilde{T..."

2012-07-19T11:07:43Z

Created page with "If $T_t$ is a strongly continuous semigroup of contraction operators on a Banach space and $\mu_t$ is a semigroup of sub-probabilistic measures on $[0, \infty)$, then \[ \tilde{T..."

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If $T_t$ is a strongly continuous semigroup of contraction operators on a Banach space and $\mu_t$ is a semigroup of sub-probabilistic measures on $[0, \infty)$, then
\[ \tilde{T}_t u = \int_{[0, \infty)} T_r u \mu_t(\mathrm d r) \]
defines another semigroup of operators $\tilde{T}_t$, which is said to be subordinate (in the sense of Bochner) to $\tilde{T}_t$.<ref name="S"/>

==Relation to Bernstein functions==
For some [[Bernstein function]] $f$,
\[ \int_{[0, \infty)} e^{-r z} \mu_t(\mathrm d r) = e^{-t f(z)} . \]
Conversely, for any Bernstein function $f$ there exists a semigroup of sub-probabilistic measures $\mu_t$ satisfying the above equality.

One often writes $\tilde{L} = f(L)$, where $-L$ and $-\tilde{L}$ are the generators of $T_t$ and $\tilde{T}_t$, respectively. This notation agrees with spectral-theoretic definition of $f(L)$ if the underlying Banach space is a Hilbert space.

==References==
{{reflist|refs=
<ref name="S">{{Citation | last1=Schilling | first1=R. | title=Subordination in the sense of Bochner and a related functional calculus | year=1998 | url=http://www.austms.org.au/Publ/Jamsa/V64P3/abs/p86/ | journal=J. Aust. Math. Soc. Ser. A | volume=64 | pages=368–396}}</ref>
}}

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Subordination - Revision history

imported>Mateusz: added "Functional calculus"

imported>Mateusz: Created page with "If $T_t$ is a strongly continuous semigroup of contraction operators on a Banach space and $\mu_t$ is a semigroup of sub-probabilistic measures on $[0, \infty)$, then \[ \tilde{T..."