All Questions
62 questions
13
votes
2
answers
552
views
Existence of closed operators with arbitrary dense domain of a given Banach space
Consider any Banach space $X$, and let $Y$ be any dense subspace, then does it necessarily exist a closed linear operator $T$ defined on $X$, such that the domain of $T$ is exactly $Y$, i.e., $D(T)=Y$?...
- 879
11
votes
3
answers
445
views
Does the generator of a 1-parameter group of Banach space isometries know which elements are entire?
Let $X$ be a complex Banach space. Let $(\sigma_t)_{t \in \mathbb{R}}$ be a 1-parameter group of linear isometries of $X$ which is strongly continuous i.e. $t \mapsto \sigma_t(x)$ is continuous for ...
- 662
11
votes
0
answers
344
views
Tauberian Theorem for 1-parameter groups of operators
The Wiener Tauberian Theorem gives condition on an $f\in L^1(\mathbb{R})$ such that the "induced 1-parameter family" $\{T_b(f)\}_{b\in \mathbb{R}}$ has a dense span in $L^1(\mathbb{R})$; ...
- 5,405
11
votes
0
answers
529
views
Contraction semigroup on Hilbert space
I'd like to know whether a certain unbounded operator on a Hilbert space is the generator of a strongly continuous contraction semigroup.
(Such operators are known as maximally dissipative operators.)
...
- 43.2k
9
votes
1
answer
202
views
Literature request: Schatten class difference of semigroups
Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily self-adjoint) such that $A, A+B$ are generators of strongly continuous one parameter semigroups $e^{-tA},e^{-t(A+B)}$...
8
votes
1
answer
747
views
Strongly continuous semigroups that cannot be contractions
Let $X$ be a Banach space, and $(P_t)_{t \ge 0}$ a strongly continuous semigroup of bounded operators on $X$. Using the uniform boundedness principle, it's simple to prove that there are constants $M,...
- 29.8k
8
votes
1
answer
261
views
A generalisation of $C_0$-semigroups
A $C_0$-semigroup is a strongly continuous family $\{T(t)\}$ of bounded linear operators on a normed space $X$, indexed in $\mathbb R_+$ and with two additional properties that make it look like an ...
- 3,388
7
votes
1
answer
593
views
Fractional powers of an operator
What is the large class of operators for which one can define fractional powers? For example, we can consider an operator $A: D(A) \subset X \rightarrow X$, generator of an analytic semigroup on a ...
- 395
7
votes
1
answer
343
views
Is every $C_0$ semigroup on a Hilbert space automatically a $C_0$ group on a larger space?
Let $\{T(t),t\ge 0\}$ be a $C_0$ semigroup on a Hilbert space $X$, does that exist a larger Hilbert space $Y$ such that $X\subset Y$, and $T(t)$ extend to a $C_0$ group $T'(t)$ (so $t<0$ make ...
- 879
6
votes
2
answers
1k
views
The contractivity of the heat semigroup in $L^p$ spaces
Let $M$ be a Riemannian manifold. By functional calculus, it is immediate to show that the heat semigroup is a contraction in $L^2(M)$. I can also show that it is a contraction in any $L^p(M)$ with $p ...
- 5,407
6
votes
1
answer
364
views
A question about uniformly bounded semigroups
Let $A$ be an unbounded linear operator of domain $D(A)$ defined on a Banach space $X$. Suppose that $A$ generates a $C_0$-semigroup $T(t)$ which is uniformly bounded. I would like to know if there ...
5
votes
1
answer
456
views
The Bochner integral about a semigroup of bounded linear operators on a Banach space
Let $T(t)$ be a semigroup of bounded linear operators on a Banach space $X$. When does the following hold
$$
\int_0^t T(s)x ds = \Big(\int_0^t T(s) ds\Big)x, \quad x \in X \, ,
$$
where $ t \in (0,1)$?...
- 51
5
votes
1
answer
475
views
Is irreducibility sufficient for uniqueness of invariant distribution for a Feller semigroup?
Let $(T_t)$ be a strongly continuous semigroup of positive operators on $C(K)$, where $K$ is a compact space. Assume also that $T_t1 =1 $ for every $t\geq 0$.
(This is also called a Feller semigroup.)
...
- 448
5
votes
0
answers
133
views
Series representation for unbounded perturbations of semigroup generators
Let $A$ generate an analytic $C_0$-semigroup on a Banach space $X$ and $B$ be a relatively compact perturbation, i.e., $B$ is compact as an operator from $D(A)$ (with the graph norm) to $X$. Then $A+B$...
- 3,388
4
votes
1
answer
156
views
approximation of a Feller semi-group with the infinitesimal generator
Let $T_t$ a Feller semigroup (see this) and let $(A,D(A))$ its infinitesimal generator.
If A is a bounded operator it is easy to show that the Feller semi-group is $e^{tA}$.
Is this formula always ...
- 293
4
votes
1
answer
113
views
Isomorphic generators
Given the two generators $(A,D(A))$ and $(B,D(B))$ of two $C_0$-semigroups on $X$ and $Y$ ( Banach spaces), respectively. We assume that there exists an isomorphism $V:D(A)\longrightarrow D(B)$ such ...
- 299
4
votes
1
answer
498
views
Generator of a $C_0$-semigroup restricted to a subspace
Suppose we have a decreasing filtration of Banach spaces $(E_t)_{t\geq0}$, inclusions $V_{s+t,t}:E_{s+t}\to E_t$ and projections $P_{t,s+t}:E_t\to E_{s+t}$ such that $P_{t,s+t}V_{s+t,t}=I_{E_{s+t}}$. ...
- 1,411
4
votes
1
answer
268
views
Restriction of a semigroup to a form domain
Say, we have a Hilbert space $H$ with a semibounded self-adjoint
operator $A:D(A)\to H$ generating a strongly continuous semigroup
$T(t):H\to H$. Is it possible to restrict $T(t)$ to a form domain of ...
- 263
4
votes
1
answer
655
views
Generator of Laplace operator as analytic semigroup on $L^1(\mathbb{R}^n)$
The Laplace operator is the generator of an analytic semigroup on $L^p(\mathbb R^n)$
for $1 < p < \infty$. Is the same true for $L^1(\mathbb R^n)$? If it is, could someone give a reference? The ...
- 271
4
votes
1
answer
259
views
$L^2$-valued integral as parameter integral
Setting
Let us regard the Hilbert space $L^2(0,1)$ and the $C_0$-semigroup $(T(t))_{t\geq 0}$ defined by
$$
T(t):\left\{
\begin{array}{rml}
L^2(0,1) & \to & L^2(0,1), \\
[f]_{\sim} &\...
3
votes
2
answers
1k
views
Lecture notes on semigroup theory for linear evolution equations
I am reading (or trying to read :)) "One parameter semigroups for Linear Evolution equations" by Klaus-Jochen Engel and Rainer Nagel. I was wondering if someone was aware of a good set of lecture ...
- 63
3
votes
2
answers
534
views
On exponential formula for $C_0$ semigroups
Let $T(t),t\geq 0$, be a $C_0$-semigroup on a Banach space $X$. If $A$ is the infinitesimal generator of $T(t),t\geq 0$, then
$$T(t)x=\lim_{n\infty}(I-\frac{t}{n}A)^{-n}x$$
for every $x \in X, t\geq ...
3
votes
1
answer
291
views
Generator of a generated $C_0$ semigroup
Consider a $C_0$-semigroup $S_t:\mathscr{B(H)} \to \mathscr{B(H)}$ with generator $U$. Now define $P_t:\mathscr{B_1(H)} \to \mathscr{B_1(H)}$ where $P_t(\rho)=S_t\rho S_t^*$. How can I prove $P_t$ ...
- 71
3
votes
2
answers
147
views
Lumer-Phillips-type theorem for non-autonomous evolutions
The classical Lumer-Phillips theorem characterizes the generators of contraction semigroups. I am looking for a similar characterization or at least a sufficient condition for a family of unbounded, ...
- 300
3
votes
1
answer
567
views
Integral representation of the resolvent of a semigroup
Let $T(t)$ be a $C_{0}$-semigroup with the generator $A$. Now, does the so called integral representation of the resolvent
$$
(\lambda - A)^{-1} = \int_{0}^{\infty} e^{-t\lambda}T(t) dt
$$
hold for ...
- 31
3
votes
1
answer
940
views
Fractional power of operators in $C_0$-semigroup
Let $X$ be Banach space, and $\{Z(t)\}_{t\geq 0}\subseteq B(X)$ be the $C_0$-semigroup of operators defined on $X$. Moreover, let $A$ be the infinitesimal generator of $\{Z(t)\}_{t\geq 0}$. A ...
- 153
3
votes
1
answer
439
views
Strong continuity of the Ornstein-Uhlenbeck operator
It's well known that the Ornstein-Uhlenbeck semigroup defined by
$$
P_tf(x)=\int_{\mathbb{R}}f\left(xe^{-t}+\sqrt{1-e^{-2t}}z\right)\frac{e^{-z^2/2}}{\sqrt{2\pi}}\,dz
$$
is not strongly continuous on ...
- 617
3
votes
1
answer
126
views
A sufficient condition for two semigroups to be norm equivalent?
Consider two densely defined, strictly positive, self-adjoint operators $A$ and $B$ with the following property
$$\|A^k x\| \simeq \|B^k x\|, \quad\forall x \in D(A^k)= D(B^k),$$
for $k=1,2,\cdots, M$,...
- 319
3
votes
0
answers
143
views
Extrapolated Integral operator (compactness)
I am studying the compactness of some convolution operators. Let the convolution with extrapolation
$$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$
Here $T(\cdot)$ is a ...
- 299
3
votes
0
answers
68
views
A strange convergence for a semigroup of operators
I am reading B. Simon's "Kato's inequality and the comparison of semigroups", and I am having troubles understanding a part of the proof of Theorem 1 therein, that goes as follows:
Let $A,B$ ...
- 5,407
2
votes
1
answer
185
views
Koopman operators on $L^p(X)$
On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ ...
2
votes
1
answer
139
views
Domain of the infinitesimal generator of a composition $C_0$-semigroup
In the paper [1] the following $C_0$-group is presented,
$$
T(t)f(x) = f(e^{-t} x) , \quad x \in (0,\infty) \quad f \in E
$$
where $E$ is an ($L^1,L^\infty$)-interpolation space. In mi case, I'm just ...
2
votes
1
answer
249
views
Positive semigroups and convex operator
Let $\{Z(t)\}_{t\geq 0}$ be a strongly continuous positive semigroup on a Banach lattice $V$ (endowed with ordering $\leq$). Let $\phi:V\rightarrow V$ be a convex operator. I want to prove that $$\phi(...
- 153
2
votes
1
answer
264
views
Bounded-pointwise continuity of Markov operators / semigroups
Let $B_b(E)$ be the space of bounded measurable functions on some Polish space $E$ endowed with the supremum norm. It seems quite classical that Markov semigroups $P_t:B_b(E)\to B_b(E)$ are in one to ...
- 59
2
votes
1
answer
181
views
Estimate of semigroup with dual norm?
Consider a semigroup $(T(t))_{t\in\mathbb{R}^+}$ generated by a densely defined strictly positive symmetric linear operator $A: D(A) \subset X \to X$, where $X$ is a Banach space with norm $\|\cdot\|$....
- 319
2
votes
0
answers
66
views
interchange of integrals and semigroup without the semigroup being an integral operator
In Cazenave's book: BREZIS, HAIM.; CAZENAVE, T. Nonlinear evolution equations. IM-UFRJ, Rio, v. 1, p. 994, 1994. The following corollary appears
The formula (1.5.2) is Duhamel formula:
$$u(t) = T(t)u(...
- 677
2
votes
0
answers
101
views
Strongly continuous semigroups on weighted $\ell^1$ space
Let $x=(x_i)$ be a sequence in $\ell^1$ such that all $x_i>0.$
Let $T(t):\ell^1 \rightarrow \ell^1$ be a strongly continuous semigroup of, i.e. $t \mapsto T(t)y$ is continuous for every $y \in \ell^...
- 536
2
votes
0
answers
55
views
$\|(A_n-z)^{-1} - (A-z)^{-1}\|\to 0\;\Rightarrow\; \|e^{-tA_n}-e^{-tA}\|\to 0$ for general $C_0$ semigroups?
In short, the question is whether norm-resolvent convergence implies operator-norm convergence of the assocoated semigroups. More specifically, assume the following:
The $A_n$ generate contraction ...
- 241
2
votes
0
answers
149
views
Projection semigroup of an isolated eigenvalue
I'm currently working with a paper and I don't get something there. Let $A$ be a closed operator on a Banach space $X$ and $\lambda \in \sigma(A)$ an isolated eigenvalue, i.e. there is a $r > 0$ ...
- 381
2
votes
0
answers
116
views
Factorization of the Fokker-Planck semigroup
I have posted this question on stackexchange over four month ago, but didn't get an answer: "In the classical theory of Markov processes, the Fokker-Planck semigroup $\{T_t:t\ge 0\}$ can be factored ...
2
votes
0
answers
144
views
Is logarithmic convexity of the heat kernel with complex time a general fact?
Suppose $H$ is a non-negative self-adjoint operator acting on $L^2(\mathbb{R}^n)$, and generates an analytic semigroup with kernel satisfying a Gaussian upper bound, i.e., if we denote $K(t,x,y)$ the ...
- 879
2
votes
0
answers
119
views
Semigroups on Banach lattice
Let $\{Z(t)\}_{t\geq 0}$ be a semigroup of positive operators on a Banach lattice $X$. I want to show that
$$\int_0^1[Z(1-s)+Z(1+s)]fds-f\in X_+,\quad f\in X_+$$
Where $X_+$ denotes the positive ...
- 55
1
vote
1
answer
430
views
Generator of an analytic semigroup of operators
I have an operator of the following form:
$$ A = \begin{bmatrix} 0 & h_1 & h_2 \\ 0 & \Delta& h_3 \\ 0 & 0 & h_4 \end{bmatrix} $$
which results from a coupled PDE-ODE system ...
- 11
1
vote
2
answers
478
views
Trotter-Kato approximation theorem for uniformly continuous approximants
Let
$E$ be a $\mathbb R$-Banach space
$(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be strongly continuous contraction semigroups on $E$ with generators $(\mathcal D(A_n),A_n)$ and $(\mathcal D(A),A)$, ...
- 167
1
vote
1
answer
292
views
A property of one-parameter groups of operators
Let $X$ be a Banach space. We consider the evolution equation:
$$x'(t)=Ax(t), \ \ \ \ \ \ \ t\in \mathbb{R},$$
where $A$ is a bounded operator.
I know that if $X=\mathbb{R^n}$ and $A$ is a matrix, ...
- 121
1
vote
2
answers
280
views
How can I prove that $(I+\lambda G)$ is invertible, where $G$ is the Green function of an elliptic operator?
How can I prove that the operator $$(I+\lambda G)$$ is invertible, where $\lambda >0$ and $G$ is the Green function of an elliptic operator $A$ in a bounded domain $\Omega$?
$\Omega$ can be very ...
- 131
1
vote
1
answer
56
views
Stability of densly defined $C_{0}$-semigroup
Let $(S(t))_{t \geq 0}$ be a $C_{0}$-semigroup on $H$ where $H$ is a Hilbert space. Suppose that $(S(t))_{t \geq 0}$ satisfies the following estimate on a dense subspace on $H$
$$||S(t)x||_H \leq e^{-...
- 617
1
vote
1
answer
284
views
Reformulation of the classical Navier-Stokes equation as a semilinear evolution equation and corresponding mild solutions
Let
$d\in\mathbb N$
$\lambda^d$ denote the Lebesgue measure on $\mathbb R^d$
$\Lambda\subseteq\mathbb R^d$ be open
$\mathcal V:=\left\{v\in C_c^\infty(\Lambda)^d:\nabla\cdot v=0\right\}$, $$V:=\...
- 167
1
vote
1
answer
96
views
Does an evolution family commute with the right shift semigroup?
Let $X$ be a Banach space and
$$\mathrm C_0(\mathbb R,X):=\{f\colon\mathbb R\to X\colon f \text{ is continuous and } \lim\limits_{|t|\to\infty}f(t)=0\}$$
normed by
$$\|f\|:=\sup\limits_{t\in\mathbb R}\...
- 33
1
vote
1
answer
253
views
Power of an infinitesimal generator of a $C_0$-semigroup in a Banach space
Let $A$ be the infinitesimal generator of a $C_0$-semigroup of linear operators in a Banach space. Let $n$ be a positive integer, $n\geq2$. Is $A^n$ closed?
Here (setting $A^1$ $:=$ $A$, and ...
- 31