All Questions
Tagged with operator-theory measure-theory
223
questions
1
vote
1
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60
views
Proving an Inequality in Hilbert Space: $\int_0^1 \chi_{[t,\infty)}(T) dt \le T$ for $T\ge 0$
Let $\mathcal H$ be a Hilbert space and $\mathcal B(\mathcal H)$ denotes the set of all bounded operators on $\mathcal H$. An element $T\in \mathcal B(\mathcal H)$ is positive, we write $T\ge 0$ if $\...
- 127
1
vote
0
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33
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Kraus operators
Suppose we have a POVM given by the family of positive, hermitian operators $\{E_i\}_{i\in I} \in \mathcal{H}$.
From the Neimark dilation theorem we know that the given POVM can be obtained from ...
- 75
-1
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0
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14
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Bergman projection maps $L^q \left (\mathbb D^2 \right )$ boundedly onto $\mathbb A^q \left (\mathbb D^2 \right )$ for any $q \geq 2.$
Let $\mathbb A^2 \left (\mathbb D^2 \right )$ be the Bergman space consisting of square integrable holomorphic functions on $\mathbb D^2$ and $\mathbb P : L^2 \left (\mathbb D^2 \right ) \...
- 2,612
4
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1
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116
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On the compactness of the square of a finite double norm integration operator $T$ on $L^1 (\mu)$.
Good morning everyone,
I have read in S. P. Eveson, Compactness Criteria for Integral Operators in L∞ and L1 Spaces, Proceedings of the American Mathematical Society 123, 1995, 3709-3716 :
"If $(...
2
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1
answer
74
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$7.24$, Real and Complex analysis, W. Rudin, Case 2.
These definitions are necessary:
There is the theorem:
If
$(a)$ $V$ is open in $R^{k}$.
$(b)$ $T : V \to R^{k}$ is continuous, and
$(c)$ $T$ is differentiable at some point $x \in V$, then $$ \lim_{r ...
- 1,020
2
votes
1
answer
70
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Papa Rudin $7.24$ Theorem,
There are some necessary definitions for the theorem:
There is the theorem:
If
$(a)$ $V$ is open in $R^{k}$.
$(b)$ $T : V \to R^{k}$ is continuous, and
$(c)$ $T$ is differentiable at some point $x \...
- 1,020
0
votes
0
answers
67
views
Some questions about integration and operator theory.
As we all know there are multiple integral operators which all basically do the same thing in various contexts. I am talking about operators like the Lebegues integral, Riemann integral and more ...
- 103
8
votes
0
answers
152
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Concentration of measure on spheres with respect to a unitary of trace approximately zero
This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-dimensional Hilbert space. First, some context:
We note that, by ...
- 9,898
2
votes
1
answer
94
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Finding dual operator - applying Fubinis theorem
For X=Y=$C([0,1])$ find the dual operator $T^*$ of $Tf(t)=\int_0^tf(s)ds$
Using the Riesz representation theorem we get $$\int_0^1f(s)d\nu(s)=T_\mu^*(f)=\int_0^1(\int_0^tf(s)ds)\;d\mu(t).$$
Now I'd ...
- 143
1
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1
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67
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How can I prove, that my constructed spectral measure $E(\Delta)$ is an orthogonal projection?
Definitions
Representation
Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$.
We call $\...
- 41
0
votes
0
answers
78
views
If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?
Definitions
Representation
Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$.
We call $\...
- 41
1
vote
0
answers
25
views
References for motivating the definition of a transfer operator
I am looking for references that motivative the following definition of a transfer operator from Wikipedia:
Let $X$ be an arbitrary set and $f: X\to X$. Then the transfer operator $\mathcal{L}$ is ...
- 1,221
0
votes
1
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57
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In a Banach Space, when is $\sum |T(x_n)| < \infty$ for every sequence where $\sum x_i$ converges?
Given a Banach space $X$, and an operator $T \in X^*$, when does it hold that
$$\sum_{i=1}^\infty |T(x_n)| < \infty$$
for every sequence $\{x_n\}$ in $X$ such that $\sum_1^\infty x_n$ converges? As ...
- 1,307
1
vote
1
answer
51
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How can we show this property of a symmetric integral operator?
Let
$D\subseteq\mathbb R^d$ be bounded and open;
$H:=L^2(D)$
$Q\in\mathfrak L(H)$ be nonnegative and self-adjoint with finite trace and hence $$Qf_n=\lambda_nf_n\tag1$$ for some orthonormal basis $(...
- 13.9k
3
votes
1
answer
187
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Supremum of a bounded set of real-valued functions in $L^\infty(X)$
Let $(X,\Sigma,\mu)$ be a measure space and consider the algebra $L^\infty(X)$. The real-valued functions carry a canonical order, where $g\leq f$ if $g(x)\leq f(x)$ a.e. The following result holds:
...
- 209k