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Results tagged with banach-spaces
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user 12898
A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.
9
votes
Dual of Banach-valued $L^p$
Diestel-Uhl, Vector measures, Section IV.1., Theorem 1:
Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, $1\leq p<\infty$ and $\frac{1}{p}+\frac{1}{p'}=1$.
The dual of $L^p(\mu;X)$ is $L^{p'}( …
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1
vote
Radon-Nikodým property of $\ell^\infty$
See Proposition 1.2.9 in Arendt-Batty-Hieber-Neubrander. (I look at the first edition now)
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6
votes
Accepted
Reference request: completion of Banach norm on sum
You find it in the manuscript of Alessandra Lunardi on Interpolation Theory. A free version is available here: http://people.dmi.unipr.it/alessandra.lunardi/LectureNotes/SNS1999.pdf
In these notes the …
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6
votes
Accepted
A question about uniformly bounded semigroups
If I understand your question correctly, this would mean that the function $t\mapsto T(t)x$ is Lipschitz continuous, which is equivalent for $x$ to be in the Favard space $\text{Fav}(A)$$. See for exa …
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12
votes
1
answer
1k
views
Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$
Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure …
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11
votes
Accepted
Strongly continuous semigroups that cannot be contractions
There are many examples constructed with weighted shifts. The following is a Hilbert space example.
Let us consider the Hilbert space $L^2\big((0,1),\mu\big)$, where $\mu$ denotes the measure defined …
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6
votes
Accepted
Does the generator of a 1-parameter group of Banach space isometries know which elements are...
If I am not mistaken, then this is connected to the notion of analytic vectors and related object.
If I understand your question correctly, it is answered in this paper of Chernoff.
Further, for gro …
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4
votes
Accepted
Reference for weak*-semigroup
Echoing the remark of @Bill Johnson, one possibility is van Neerven's book on adjoint semigroups.
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3
votes
Generator of a $C_0$-semigroup restricted to a subspace
I do not think such semigroups have been extensively studied.
I have seen such semigroups (and, more generally, evolution families corresponding to the non-autonomous problem) in
A. Lunardi, M. Geis …
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7
votes
The Bochner integral about a semigroup of bounded linear operators on a Banach space
Following ideas of John von Neumann,
J. von Neumann, Über einen Satz von Herrn M. H. Stone, Ann. Math. (2) 33, 567-573 (1932). ZBL0005.16402,
it can be shown that if a function satisfies the semigro …
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4
votes
Accepted
Characterization of the interpolation space $(X,D(A^\alpha))_{\theta,p}$ with semigroup $A$ ...
Yes, there is a characterization like this. See Theorems 1-3 (p.182) in
Markus Haase, MR 2183483 A functional calculus description of real interpolation spaces for sectorial operators, Studia Math. 1 …
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