All Questions
Tagged with operator-theory analysis
494
questions
0
votes
1
answer
45
views
Is my formula for this projection correct?
Let $\phi \in L^{2}(\mathbb{R}^{d})$ be fixed. Denote by $P$ the orthogonal projection onto the subspace orthogonal to $\text{span}\{\phi\}$. In other words, for $f \in L^{2}(\mathbb{R}^{d})$ set:
$$(...
- 923
2
votes
1
answer
74
views
$7.24$, Real and Complex analysis, W. Rudin, Case 2.
These definitions are necessary:
There is the theorem:
If
$(a)$ $V$ is open in $R^{k}$.
$(b)$ $T : V \to R^{k}$ is continuous, and
$(c)$ $T$ is differentiable at some point $x \in V$, then $$ \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
2
votes
1
answer
56
views
Characterization of normal Fredholm Operators
I'm working on the following problem, but I'm getting stuck on the last part. Here is the problem:
Let $T \in \mathcal{B}(H)$ be normal. Prove $T$ is Fredholm if and only if $0$ isn't a limit point ...
- 1,399
1
vote
0
answers
31
views
Boundedness of an Integral Operator on $L^p(\mu)$
I realize this question has been asked way too many times, for instance, it's this exact problem which I will put below for convivence:
Let $(X,\Omega, \mu)$ be a $\sigma$-finite measure space with $...
- 1,399
2
votes
1
answer
54
views
Examples of The Gelfand Representation Not Behaving
Currently I am trying to think of examples of Banach algebras $A$ where the Gelfand Map $\Gamma: A \longrightarrow C_0(\Omega(A))$ fails to be an isomorphism. So far, I have come up with some mundane ...
- 1,399
1
vote
1
answer
36
views
Image of the closed unit ball is weak* dense
I'm aware that there are a few posts about this question here, but none of the proofs make the last step of this argument make any sense to me.
Here is the problem statement: Let $X$ be Banach and $\...
- 1,399
2
votes
1
answer
44
views
Spectral Permanence Remark in Murphy's C*-algebras
In Murphy's $C^{*}$-algebras book, he states theorem $2.1.11$ which is that if $\mathfrak{B} \subset \mathfrak{A}$ are $C^{*}$-algebras with $\mathfrak{A}$ unital such that $1_{\mathfrak{A}} \in \...
- 1,399
0
votes
1
answer
46
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Let $f\colon H \to L(H,X)$ strongly continuous ($X,H$ Hilbert spaces). Is the adjoint function $f^*\colon H \to L(X,H)$ strongly continuous?
Let $X,H$ be Hilbert spaces and let $f\colon H \to L(H,X)$, where $L(H,X)$ is the space of linear bounded operators. In this case it is natural to define the adjoint function $f^*\colon H \to L(X,H)(=...
- 185
2
votes
0
answers
88
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Is this statement about quaternion function true?
Let $\mathbb{H}=\{q=t+xi+yj+zk:t,x,y,z\in\mathbb{R}\}$ be the quaternions and $f:\mathbb{H}\to\mathbb{H}$ be a function satisfying
$\overline{\partial_c}f=0,$
where
$$\overline{\partial_c}=\frac{1}{2}\...
0
votes
0
answers
30
views
Composition of an operator and a bounded holomorphic mapping
If $f \in \mathcal{H}^\infty(U,F)$ is a bounded holomorphic mapping from an open subset of a complex Banach space $U$ into a complex Banach space $F$, then I know that $T \circ f = 0$, for all bounded ...
- 215
0
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0
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30
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How should I interpret this operator equality?
Suppose $A$ is a fixed densely defined self-adjoint operator on a complex Hilbert space $\mathfrak{h}$.We are given a function $f \in C_{0}^{\infty}([0,\infty))$ with compact support and smooth such ...
- 923
1
vote
1
answer
116
views
Inversion/Injectivity of Symmetric Translation Operator in $L^2(\mathbb{R})$ (via Distribution Theory)
Consider an operator (T_a) acting on elements of $L^2(\mathbb{R})$, defined to symmetrically translate a function by plus and minus $a$. This operator is mathematically expressed as:
$$
(T_a f)(x) = \...
- 161
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
0
answers
20
views
Operator Monotony of composed maps
It is a common fact from Matrix Analysis (see e.g. Bhatia 1997, "Matrix Analysis") that the map $t\mapsto t^r$ is matrix and in fact operator monotone on $t\in[0,\infty)$ if and only if $r\...
- 11