Tagged Questions

-1
votes
0answers
12 views

Help with Toeplitz operators applications. [closed]

I am trying to find a physics problem which solution involves Toeplitz operators.
1
vote
1answer
43 views

Composition of $\mathrm H^p$ function with Möbius transform

Let $f:\mathbb D\rightarrow \mathbb C$ be a function in $\mathrm{H}^p$, i.e. $$\exists M>0,\text{ such that }\int_0^{2\pi}|f(re^{it})|^pdt\leq M<\infty,\forall r\in[o,1)$$ Consider a Möbius ...
1
vote
1answer
55 views

Computing an explicit solution to an integral equation via the Neumann Series.

I am hoping that someone can provide guidance for solving the integral equation $$u=f+\lambda Au$$ where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
4
votes
0answers
47 views

When are two commuting linear operators functions of each other

I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up. If we formally consider the integral operator $E ...
1
vote
0answers
31 views

Is there an algebra for divergent series summation operators?

Let $D$ denote a divergent series and let $C$ denote a convergent series. Furthermore, let $s : \{ Series \} \to \{ numbers \} $ be a regular, linear divergent series operator, which is either one of ...
2
votes
1answer
84 views

spectrum of two bounded linear operators

Suppose that L and B are bounded linear operator on H, assume $0\in \rho(L) \cap \rho(L+B)$ and that $L^{-1}$ is compact. Prove that L+B also has a compact inverse.
1
vote
1answer
39 views

Convergence in norm operator topology

I have to prove that a sequence $A(\varepsilon)$ of operators between Hilbert spaces $A(\varepsilon):H_1\to H_2$ converges, when $\varepsilon\to 0^+$, to an operator $B:H_1\to H_2$ in the uniform norm ...
2
votes
1answer
102 views

Point spectrum in Hilbert spaces

Let $H$ be a Hilbert space and and $T\in B(H)$ be normal and $\sigma_p(T)$ be the point spectrum of $T$ (i.e the set of all eigenvalues of T) and let $E$ denote the spectral measure. I'm trying to ...
1
vote
1answer
28 views

Exponential of nth order derivative

When dealing with an exponential operator of the form $e^{\vec a \cdot \vec \nabla}f(\vec x)$, I understand how this simply shifts the argument of the function by $a⃗$ . My question is what happens ...
2
votes
1answer
57 views

Self-adjoint operator and inner product

I am wondering whether there is a way to make sense of self adjointness of an operator on $C[0,1]$ without resorting to the inner product of $L^2[0,1]$. I am not referring to concrete alternative ...
2
votes
1answer
36 views

Strong convergence of multiplication operator

I am looking for a necessary and sufficient condition for a sequence of multiplication operators $T^{(k)}$ to converge to zero strongly. (i.e. $\forall x \in \mathcal{H} \quad ||T^{(k)}x - 0|| \to 0$ ...
1
vote
2answers
70 views

How to prove that two non-zero linear functionals defined on the same vector space and having the same null-space are proportional?

Let $f$ and $g$ be two non-zero linear functionals defined on a vector space $X$ such that the null-space of $f$ is equal to that of $g$. How to prove that $f$ and $g$ are proportional (i.e. one is a ...
0
votes
3answers
161 views

How to find the norm of this bounded linear functional?

Let $C^\prime[a,b]$ denote the normed space of all continuously differentiable real (or complex) valued functions defined on the closed, bounded interval $[a,b]$ in $\mathbf{R}$ with the norm defined ...
1
vote
1answer
97 views

What is the norm of this bounded linear functional?

Let $a$, $b$ be two arbitrary but fixed real numbers such that $a < b$, let $C[a,b]$ denote the normed space of all continuous real (or complex) valued functions defined on $[a,b]$ with the maximum ...
1
vote
1answer
70 views

What are the range and the norm of this bounded linear operator?

Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
1
vote
1answer
78 views

How to find the range and inverse of this linear operator?

Given $T \colon C[0,1] \to C[0,1]$ defined by $$Tx(t):= \int_0^t x(r) dr$$ for each $t\in [0,1]$, where $C[0,1]$ is the normed space of continuous real-valued (or complex-valued) functions defined on ...
2
votes
3answers
97 views

Norm of bounded operator on a complex Hilbert space.

It is fairly easy to show that for a bounded linear operator $T$ on a Hilbert space $H$ $$||T||=\sup_{||x||=1,||y||=1}|\langle y, Tx \rangle |.$$ If $H$ is a complex Hilbert space, can you show that ...
1
vote
0answers
53 views

If limit of $f(n)$ is zero then the operator is compact

I want to prove the following: Suppose $\mathfrak{H}$ is the Hilbertspace $l^2(\Bbb{N})$ and $T_f$ the multiplication operator on $\mathfrak{H}$, thus $T_f\psi=f\psi$ for ...
3
votes
2answers
116 views

Proof that certain operators are compact

I want to examine which of the following operators $T \colon C[0,1] \to C[0,1]$. are compact, by some I think I got the argument, but others I have no idea. a) $Tx(t) = x(t^2)$ Guess it is ...
1
vote
1answer
120 views

Exponential operator on a Hilbert space

Let $T$ be a linear operator from $H$ to itself. If we define $\exp(T)=\sum_{n=0}^\infty \frac{T^n}{n!}$ then how do we prove the function $f(\lambda)=exp(\lambda T)$ for $\lambda\in\mathbb{C}$ is ...
2
votes
1answer
91 views

Norm of operator $g\mapsto \int fg$

Let $T_f:(C([a,b],\mathbb C), \lVert \cdot \rVert_1) \to \mathbb C$ with $g\mapsto \int_a^b f(x)g(x) dx$ for any given $f\in C([a,b],\mathbb C)$. I have to find the norm of $T_f$. I started with: ...
3
votes
1answer
178 views

Reflexivity of a Banach space

I've run into a few problems in which reflexivity of a Banach space is given as a hypothesis. These problems are sometimes of the type where the banach space is specific/concrete, and sometimes it is ...
0
votes
1answer
70 views

Weak analyticity vs. Strong Analyticity

Let $X$ be a (complex) banach space, $U$ be an open subset of $\mathbb{C}$ and $f: U \to X$ be a function that is completely arbitrary except that it satisfies the property that for any continuous ...
4
votes
2answers
153 views

Help for Divergence operator

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple. Can some one tell me some reference to study about the invertibility of Divergence operator ...
0
votes
1answer
64 views

Inversion in a unital C* algebra

Let's say that in a unital C* algebra, we have $b \geq a \geq 0$ and $a$ is invertible. Then $b$ is also invertible. Can we conclude that $a^{-1} \geq b^{-1}$? If so, why? Can any related ...
1
vote
1answer
113 views

A specific example of the GNS construction

In an introduction to the GNS construction, I'm told that the GNS construction is a generalization of the way that $L^{\infty} (X, \mu)$ has a representation on $L^2$ where $\mu$ is a measure on $X$. ...
3
votes
1answer
456 views

Compactness and boundedness of integral operator

I got some trouble with my homework question : Let $B$ be the unit ball in $\mathbb{R}^d$, and let $T$ be an integral operatpor on $L^2(B)$ with kernel $K(x,y)$. Suppose that $\sup_x \int_B ...
3
votes
1answer
98 views

Operator norm of the sum of a finite collection of bounded linear operator

I recently got some difficulty with my homework question. The question is: Let $T_1,\dots,T_N$ be a finite collection of bounded linear operators on a hilbert space $H$, each of operator norm $\le ...
6
votes
1answer
89 views

A Marcinkiewicz approach

The problem was to prove the following that the operator $$Tf(x)=\int_{\mathbb{R}^N}\frac{f(y)}{|x-y|^\alpha}dy$$ Is continuous from $$L^1 \to \ L_\mathrm{Weak}^{p}$$ where $0<\alpha<N$ and ...
2
votes
0answers
122 views

Eigenfunctions/Invariance of generic convolution operators

Suppose we are given a convolution operator $$ \mathcal{K}[f\,](t):=\int K(t-s)f(s)ds $$ acting on $f\in H_1$ where $H_1$ is a vector space with orthonormal basis $\{\phi_n(t)\}_{n=0}^{N-1}$. If ...
0
votes
1answer
104 views

Two questions from Dixmier's book on Von Neumann algebras

It seems something is going wrong with the preview I linked in some of my previous questions, so I will just type out the question. I am having trouble with Dixmier's proof of Corollary 5 on p. 46. ...
0
votes
0answers
83 views

weak closures of ideals [duplicate]

Possible Duplicate: Two questions from Dixmier's book on Von Neumann algebras On p. 46-47 in Dixmier's book on Von Neumann Algebras, which I just realized can be accessed through this ...
3
votes
0answers
106 views

Two questions about ultraweak and ultrastrong topology from Dixmier

You could reference Dixmier's book on Von Neumann Algebras p.42 Theorem 1 and its proof to know the entirety of the context. Otherwise, the most relevant things are below: Let $M$ be an ultraweakly ...
0
votes
0answers
92 views

The bipolar theorem from functional analysis

I am referring to this source: http://www.math.unl.edu/~s-bbockel1/929/node6.html. How can I show the other bipolar theorem where the polar of a prepolar is the closed convex balanced hull? Is it ...
0
votes
0answers
89 views

polars in functional analysis in Dixmier

On page 39 of Dixmier's text on Von Neumann Algebras, he argues for Lemma 1, in which he tries to see that $\theta(L_1)=E_1$ using an argument about polars from functional analysis. I was hoping ...
4
votes
2answers
122 views

Duals via a Bilinear map

Let $E$ and $F$ be normed vector spaces. Then if $B$ is a bounded bilinear form on $E \times F$ then every $y \in F$ defines a bounded linear functional $f_y$ where $f_y(x)=B(x, y) \forall x \in E$. ...
1
vote
0answers
163 views

Convergence of net sums of complex numbers, as well as operators

I have some questions concerning convergence of sums where the summands are complex number, although the real motivation of my question comes from Von Neumann algebras where sometimes the summands are ...
0
votes
0answers
109 views

Interchanging Strong Operator convergent sums

In the book on operator algebras by Stratila and Zsido, they discuss in Ch.2 the idea of taking a Hilbert space $H$ and an index set $I$ and associating to it the Hilbert space that is the direct sum ...
1
vote
2answers
107 views

Does the inequality $0\leq a\leq b$ in a C*-algebra imply $\|a\|\leq\|b\|$?

In relation to this question of mine: C* algebra inequalities I am wondering if it is true that if $0\leq a \leq b$ in a C* algebra, does one have $||a||\leq||b||$? If you need the C* algebra to be ...
2
votes
1answer
181 views

Tensor product of Hilbert Spaces

I am following this link under "definitions" I need to see why the suggested inner product on the pre-Hilbert space $H_1$ tensor $H_2$ is well defined. Recall that the fundamental tensors are a ...
2
votes
1answer
54 views

Classification of Type 1 factors

In the proof of this theorem, which says all of the type 1 factors (factors with minimal projections) are isomorphic to $B(\ell^2(I))$ for some $I$, I want to know a few things: The supposed ...
2
votes
1answer
101 views

Generation of Von Neumann Algebras

Suppose $M$ is a Von Neumann Algebra. (VNA) For me, these are subsets of some $B(H)$ that are $*$-algebras, containing the $1$ of $B(H)$, that are Weak Operator (WO) closed, or equivalently Strong ...
1
vote
1answer
555 views

Why call this a spectral projection?

Regarding this question, Why do spectral projections give norm approximations? I have figured out what is meant by spectral projection, and have thus found the answer as well. A spectral projection ...
4
votes
1answer
258 views

Compact multiplication operators

In class, we started talking about operators on Banach spaces after covering the Arzela-Ascoli Theorem. We defined a continuous operator $T\colon X \to Y$ to be compact if $\overline{T(B_X)}^{Y}$ is ...
8
votes
4answers
275 views

Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question... Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty ...
6
votes
1answer
205 views

Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$

Consider the fractional integro-derivative $\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi ...
1
vote
1answer
36 views

What is a semibounded polynomial on $\mathbb{R}^n$?

I am stuck with the following expression, because no google search gives an answer to my problem. Here it is: I am reading a text that states "Let a(x) be a semibounded from below polynomial on ...
6
votes
3answers
836 views

Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?

Using the Taylor expansion $$f(x+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dx^k}f(x)$$ one can formally express the sum as the linear operator $e^{a\frac{d}{dx}}$ to obtain $$f(x+a) = ...