Skip to main content

All Questions

Filter by
Sorted by
Tagged with
11 votes
3 answers
3k views

Dual space of $L^2(\mathbb{R},L^1(0,1))$?

I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures) Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
Jacob Augstine's user avatar
7 votes
1 answer
306 views

An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that $$ f^2=f $$ (that ...
limanac's user avatar
  • 452
5 votes
1 answer
2k views

definition of operator valued integral with spectral measure

I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011). There, they work on a Hilbert space $H$ and on the ...
Yul Otani's user avatar
  • 342
4 votes
1 answer
313 views

Invariant subspaces are reducing subspaces in $L^2(\mu)$; where $\mu$ is a singular measure w.r.t Lebesgue measure

I have already posted this question on math.stackexchange but didn't get any answer. I hope this is the right place to ask this question. Recently I was reading a book "Operator Function and system" ...
Timon's user avatar
  • 207
4 votes
1 answer
330 views

Uniform boundedness principle for almost surely converging sequence of operators

I'd like to do the following: I consider a separable Banach space $X$ with a probability measure $\mu$ on the Borel $\sigma$-algebra $\mathcal B(X)$. Additionally, I have a sequence of measurable, ...
Philipp Wacker's user avatar
4 votes
1 answer
2k views

Operator topologies on $L^{\infty}(X,\mu )$

Let $(X,\mu )$ be a measure space. Then, $L^2(X):=L^2(X,\mu )$ is a Hilbert space in the usual way and we may view $L^{\infty}(X):=L^{\infty}(X,\mu )$ as a subalgebra of bounded operators on $L^2(X)$ ...
Jonathan Gleason's user avatar
4 votes
1 answer
283 views

Absolutely continuity in variation of constant formula

We are talking here about the initial value problem on some Hilbert space $H$ $$y'(t)=Ay(t)+f(t), \\ y(0)=y_0 \in D(A).$$(Problem 1.13 in the reference) Then $y(t)=e^{At}y_0 + \int_0^t e^{A(t-s)}f(s) ...
Torpedo's user avatar
  • 43
3 votes
1 answer
216 views

Extension of a bilinear functional

Does any one know an example of a bilinear functional $B:C(X)\times C(Y)\to {\bf R}$ ($X$ and $Y$ are open subsets of Euclid spaces) which cannot be extended continuously to a measure $\mu:C(X\times Y)...
darko's user avatar
  • 31
3 votes
1 answer
274 views

Function square-integrable

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$ where $x_0$ is an ...
Andrea Tauber's user avatar
3 votes
0 answers
101 views

Pettis vs. Dunford integrability of operator valued functions

Given a Banach space $X$ and a measure space $(\Omega ,\mu )$, one says that a function $$ f:\Omega \to X $$ is Dunford integrable, or scalarly integrable if, for every $\varphi $ in the ...
Ruy's user avatar
  • 2,263
3 votes
0 answers
225 views

Defining a trace-class operator with a Bochner integral

I had asked this question previously on Math.StacheExchange but did not get an answer there in several months. This isn't strictly speaking research level mathematics but I hope it is sufficiently ...
Adomas Baliuka's user avatar
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 ...
Scottish Questions's user avatar
2 votes
2 answers
431 views

References for the Spectral Theorem ( Multiplication Operator Form)

Let $A_1$ and $A_2$ be two commuting self-adjoint (or normal) operators on an infinite-dimensional complex Hilbert space $E$, then there exists a measure space $(X,\mathcal{E},\mu)$, two functions $\...
Student's user avatar
  • 1,154
2 votes
1 answer
140 views

Measurability of $T \to \Pi_{ker T}$ w.r.t. SOT

I came across the following technical question, to which I could not - after some time of thinking - find an answer: Let $\mathcal{U},\mathcal{H}$ be two real (in general infinite dimensional) ...
PDEprobabilist's user avatar
2 votes
1 answer
577 views

When is the bound in Riesz-Thorin Interpolation Theorem attained?

Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem). Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...
Eusebio Gardella's user avatar
2 votes
1 answer
135 views

A non-condensing operator with a power condensing

Let $\alpha$ to be the Kuratowski measure of non-compactness, in a Banach space $E$. It's very easy to prove that $\alpha (D_1\times D_2)\leq \alpha (D_1)+\alpha (D_2)$, where $D_1$ and $D_2$ are ...
Motaka's user avatar
  • 291
2 votes
0 answers
94 views

Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces

Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
Bogdan's user avatar
  • 1,759
2 votes
1 answer
149 views

Show that $\|P(f\circ\varphi_{\lambda})-\widetilde{f}(\lambda)\|_p=\|P(f\circ\varphi_{\lambda}-\overline{P(\overline{f}\circ\varphi_{\lambda}}))\|_p.$

Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman ...
Anacardium's user avatar
2 votes
0 answers
40 views

Invariance of simple functions

Let $(A(s),D(A(s)))_{s\in\mathbb{R}}$ be a family of unbounded operators on a Banach space $X$ and $g:\mathbb{R}\rightarrow X$ be a simple function, i.e., \begin{align*} g=\sum_{i=1}^n{x_i\textbf{1}_{...
Travis Mccormick's user avatar
2 votes
0 answers
58 views

Absolute continuity of DOS measure for Schrödinger operators

Kotani theory gives roughly that for ergodic operators there is a certain equivalence between absolutely continuous spectrum and an absolutely continuous density of states measure. I would like to ...
DDriggs's user avatar
  • 21
1 vote
1 answer
151 views

Some operators on spheres

Let $S_2$ be the unit sphere in $\mathbb R^3$ equipped with normalized Haar measure. For a continuous function f and $\delta\in (-1,1)$ define $T_\delta f(x):=\int_{\{y:<x,y>=\delta\}}f(y)d_\...
A beginner mathmatician's user avatar
1 vote
1 answer
80 views

Prove that $N_1(A_1,A_2)= N_2(A_1,A_2)$ when $A_1A_2=A_2A_1$ and $A_1$ and $A_2$ are normal operators

Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$. On $\mathcal{L}(E)^2$, we have two equivalent norms: \begin{eqnarray*} N_1(A_1,A_2) &=&\sup\...
Student's user avatar
  • 1,154
1 vote
1 answer
165 views

Integral function $z(x):=\int_{Y} f(x,y)d\mu(y)$ continuous?

Let $z(x):=\int_{Y} f(x,y)d\mu(y)$ for $x \in \mathbb R$ be an integral function where $\mu$ is a finite(!) Borel measure on $Y$ and $x \mapsto f(x,y)$ is continuous for every $y.$ Moreover, we know ...
Sascha's user avatar
  • 536
1 vote
1 answer
287 views

Interpolation between $L^1$ and $L^2$ spaces

I was wondering whether the following interpolation between $L^1$ and $L^2$ spaces is true: Let $f \in \mathbb{R}^n$ be such that $$ \alpha_1:= \int_{\mathbb{R}} \left\lVert f(x_1,\cdot,....\cdot) \...
Jacob Augstine's user avatar
1 vote
0 answers
101 views

If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?

Definitions Representation Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$. We call $\...
S-F's user avatar
  • 63
1 vote
0 answers
135 views

Description of state space of $C(K,M_n)$?

Edit: closed convex hull added. I am trying to understand the state space of $C(K,M_n)=C(K)\otimes M_n$ for $K$ a compact space. My guess would be that these are the closed convex hull of states on $C(...
C-star-W-star's user avatar
0 votes
0 answers
59 views

Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?

I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
S-F's user avatar
  • 63
0 votes
0 answers
49 views

Existence of sequence of regular projections

Reading the book :Krasnosel'skii, M.A.; Pustylnik, E.I.; Sobolevskii, P.E.; Zabreiko, P.P. (1976), Integral Operators in Spaces of Summable Functions, Leyden: Noordhoff International Publishing, 520 p....
Guillermo García Sáez's user avatar