All Questions
Tagged with operator-theory real-analysis
805
questions
2
votes
1
answer
74
views
$7.24$, 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 $$ \lim_{r ...
- 1,020
2
votes
1
answer
70
views
Papa Rudin $7.24$ Theorem,
There are some necessary definitions for the theorem:
There is the theorem:
If
$(a)$ $V$ is open in $R^{k}$.
$(b)$ $T : V \to R^{k}$ is continuous, and
$(c)$ $T$ is differentiable at some point $x \...
- 1,020
0
votes
0
answers
40
views
Lower Bounds for the Operator norm of Integral Operators with square-integrable kernels
Let $k$ be a measurable function on $E\times E$ where $E \subset \mathbb{R}^2$, define the integral operator $L_k$ by
$$
L_k f(x) = \int_{E} k(x,y) f(y) dy, \qquad x \in E
$$
If $k$ is is square-...
- 6,106
2
votes
1
answer
75
views
Applying Spectral Mapping Theorem to determine if $f(T)$ is compact
Given a self-adjoint, compact operator $T$ and a continuous $f:\sigma(T)\to\mathbb{R}$, I'm trying to determine the conditions under which $f(T)$ is also compact. I know of the Spectral Mapping ...
- 529
1
vote
0
answers
42
views
Multiplication Operator Version of Spectral Theorem in a Rigged Hilbert Space
(This is a reformulation of a deleted post which didn't quite reflect the confusion i have)
Consider the setting of a Hilbert space $\mathscr{H}$. A function $f \in \mathscr{H}$ can be represented in ...
- 161
1
vote
1
answer
40
views
Showing that $\lim_{h\to 0}\frac{1}{h}e^{\mu h}\int_0^h e^{-\mu t}A(t)xdt = A(0)x$
Let $A(t)\in\mathbb{C}^{n\times n}$ be a continuous matrix-valued function in $t$ (in matrix operator norm) for $t\geq 0$ such that $\|A(t)\| \leq Ce^{b t}$ for $b \in\mathbb{R}$ and some $C\geq 1$. ...
- 1,030
2
votes
1
answer
87
views
Extending Injectivity of a Fourier Multiplier Operator from $C^\infty_c$ to $L^2$ Using Density Arguments
Suppose an operator $T$ is well-defined on $L^2(\mathbb{R})$, but its injectivity is explicitly established for functions in the subspace $C^\infty_c$. Given the dense embedding of $C^\infty_c$ in $L^...
- 505
3
votes
1
answer
267
views
Is there a closed form for the linear operator $T$ such that $T(x^n) =f(n)x^{n-1}$?
When I first learnt calculus I was so surprised to learn that there is a meaningful mathematical operator $D$ that
$$
D(x^n)= n x^{n-1}.
$$
It seemed to be a very random thing to multiply the exponent ...
- 6,620
2
votes
2
answers
114
views
Estimate norm of convolution operator
I'm trying to find the operator norm for $T: L^2([0,1])\to L^2([0,1])$, defined as $Tf(x)=\int_{[0,1]}|\sin(x-y)|^{-\alpha}f(y)dy$, where $0<\alpha<1$. Using an upper bound on $|\sin(x)|\geq |x|/...
- 529
1
vote
1
answer
116
views
Inversion/Injectivity of Symmetric Translation Operator in $L^2(\mathbb{R})$ (via Distribution Theory)
Consider an operator (T_a) acting on elements of $L^2(\mathbb{R})$, defined to symmetrically translate a function by plus and minus $a$. This operator is mathematically expressed as:
$$
(T_a f)(x) = \...
- 161
2
votes
0
answers
57
views
Is this integral operator a compact operator?
Let $D$ be the closed unit ball in $\mathbb{R}^n$. Define $$(T x)(t)=\int_{D}\frac{x(s)}{|t-s|^\alpha}{\rm d}s$$ Where $0<\alpha<n$. Then $T$ is a continuous operator on $C(D)$. Is it compact?
I'...
- 81
0
votes
0
answers
23
views
How to show that $A_h$ $\gamma$-converges to $A$ iff $\lim_{n\rightarrow \infty} \sup_{|f|_{L^2}\leq 1}\|R_h(f)-R(f)\| = 0. $
I am reading a paper by Butazzo, Dal Maso. Let $\Omega$ be a bounded open set in $\mathcal{R}^n$. Let $A_h$, $A$ $\subset \Omega$ be (quasi-)open. Consider the following two PDEs with Dirichlet ...
1
vote
0
answers
94
views
Using density argument of Schwartz functions in $L^2$ to show an operator is symmetric
The position operator $X$ is defined as multiplication by $x$, i.e.
$$(X\psi)(x):=x\psi(x).$$
We take the domain $D(X)\subset L^2(\mathbb R)$ for $X$, which we can take
$$D(X):=\{\psi \in L^2(\mathbb ...
- 5,601
1
vote
1
answer
58
views
Showing that the integral operator $Ku(x):=\int_{A}k(x,y)u(y)dy$ is a Hilbert-Schmidt operator if $k\in L^2(A^2)$ for $A\subset\mathbb{R}$
Let $A\subset\mathbb{R}$ and $k\in L^2(A^2;\mathbb{C})$. I am trying to understand how we can conclude that the integral operator $K$ defined to act on $L^2(A;\mathbb{C})$ by $Ku(x) := \int_Ak(x,y)u(y)...
- 1,221
0
votes
2
answers
65
views
Proof Position Operator Is Dense
This is an exercise from my last homework sheet, proofing that $P$ is unbounded and self-adjoint was clear, however I'm having trouble proofing that $P$ is densely defined.
How my Instructor solved ...
- 389