Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.
12
votes
1answer
557 views
How to prove that an operator is compact?
Consider $T\colon\ell^2\to\ell^2$ an operator such that
$Te_k=\lambda_k e_k$ with $\lambda_k\to 0$ as $k \to \infty$ how to prove that it is compact?
12
votes
3answers
611 views
Compactness of a bounded operator $T\colon c_0 \to \ell^1$
Pitt Theorem says that any bounded linear operator $T\colon \ell^r \to \ell^p$, $1 \leq p < r < \infty$, or $T\colon c_0 \to \ell^p$ is compact.
I know how to prove this in case $\ell^r \to ...
2
votes
1answer
439 views
Question about Fredholm operator
$X,Y$ are Banach spaces and $A\in B(X,Y)$ is a Fredholm operator (that is, the dimensions of ker($A$) and coker($A$) are both finite), then are closed linear subspaces ker($A$) and Im($A$) ...
6
votes
1answer
310 views
Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $
Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $.
Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
17
votes
2answers
506 views
Compact sets as point spectrum of a bounded operator
It is well known that if $K$ is any compact set in $\mathbb{C}$, then there exist a bounded linear operator $T:l_2\to l_2$ such that $\sigma(T)=K$. My questions are:
Q1) Does there exist $T$, a ...
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) = ...
5
votes
1answer
723 views
Easy Proof Adjoint(Compact)=Compact
I am looking for an easy proof that the adjoint of a compact operator on a Hilbert space is again compact. This makes the big characterization theorem for compact operators (i.e. compact iff image of ...
2
votes
1answer
59 views
Graph of symmetric linear map is closed
A homework problem:
Let $H$ be a Hilbert space.
Let $T:H\rightarrow H$ be a symmetric linear map ($\langle Tx,y\rangle=\langle x,Ty\rangle$).
Show that $S$ is bounded.
My attempt: I'd ...
2
votes
1answer
206 views
Hellinger-Toeplitz theorem use principle of uniform boundedness
Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all ...
13
votes
1answer
474 views
Why do zeta regularization and path integrals agree on functional determinants?
When looking up the functional determinant on Wikipedia, a reader is treated to two possible definitions of the functional determinant, and their agreement is trivial in finite dimensions.
The first ...
5
votes
1answer
79 views
local convexity of $L_p$ spaces
wiki says The spaces $L_p([0, 1])$ for $0 < p < 1$ are equipped with the F-norm
they are not locally convex, since the only convex neighborhood of zero is the whole space
Why is this so? ...
8
votes
1answer
433 views
Closure of the invertible operators on a Banach space
Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ ...
9
votes
1answer
211 views
Every Hilbert space operator is a combination of projections
I am reading a paper on Hilbert space operators, in which the authors used a surprising result
Every $X\in\mathcal{B}(\mathcal{H})$ is a finite linear combination of orthogonal projections.
The ...
7
votes
2answers
199 views
Proof that operator is compact
Prove that the operator $T:\ell^1\rightarrow\ell^1$ which maps $x=(x_1,x_2,\dots)$ to $\left(x_1,\frac{x_2}{2},\frac{x_3}{3},\dots\right)$ is compact.
For an arbitrary sequence $x^{(N)}\in\ell^1$ ...
6
votes
1answer
166 views
Questions about a PDE: $-u_{txx}+u_{t}=u_{x}$, $u(x,0)=u_0(x)$, $x,t\in{\bf R}$
Consider the BBM equation:
$-u_{txx}+u_{t}=u_{x}$, $u(x,0)=u_0(x)$, $x,t\in{\bf R}$.
One may rewrite this equation as following
$u_t=((I-A)^{-1}\partial_x)u$ where $Au=u_{xx}$ if $(I-A)^{-1}$ ...
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 ...
3
votes
1answer
199 views
Operators on $C([0,1])$ that is compact or not.
For $f\in C([0,1])$ set
$$Hf(x) = \frac{1}{x}\int_0^x f(t)dt.$$
a) Show that $H$ is a bounded operator from $C([0,1])$ into itself which is not compact.
b) From a) it follows that $H$ induces a ...
2
votes
1answer
94 views
Hilbert's Inequality
Could you help me to show the following:
The operator
$$
T(f)(x) = \int _0^\infty \frac{f(y)}{x+y}dy
$$
satisfies
$$\Vert T(f)\Vert_p \leq C_p \Vert f\Vert_p
$$
for $1 <p< \infty$ where
...
1
vote
1answer
113 views
Normal operators in Hilbert spaces
Let $H$ be a separable Hilbert space and let $T:H\to H$ be a continues linear map such that there exists an orthonormal basis of $H$ that consists of the eigenvectors of $T$. Show that $T$ is normal. ...
1
vote
1answer
205 views
Compact operator? self adjoint operator? Stirling's formula
Define a pair of operators $S\colon\ell^2 \rightarrow \ell^2$ and $M\colon\ell^2 \to\ell^2$ as follows: $$S(x_1,x_2,x_3,x_4,\ldots)=(0,x_1,x_2,x_3,\ldots) $$ and ...
5
votes
1answer
174 views
Bounded operator and Compactness problem
Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator.
a) Let $x\in [a,b]$. Show that there is a ...
4
votes
1answer
226 views
Sum of Closed Operators Closable?
Let $A$ and $B$ be closed operators on a (separable complex) Hilbert space with dense domains $D(A)$ and $D(B)$ respecitvely. Then, we may define the operator $A+B$ on $D(A)\cap D(B)$. In general, ...
2
votes
1answer
170 views
Different types of continuity for operators on Hilbert spaces
In chapter one of K-theory and $C^*$-algebras, a Friendly Approach, the author gives a very brief discussion about several types of continuity of operators between Hilbert spaces.
Let $T: ...
2
votes
1answer
221 views
A compact operator is completely continuous.
I have a question.
If $X$ and $Y$ are Banach spaces, we have to prove that a compact linear operator is completely continuous.
A mapping $T \colon X \to Y$ is called completely continuous, if it maps ...
2
votes
1answer
183 views
Weakly compact operators on $\ell_1$
Is the following assertion true/known?
Let $V$ be a Banach space and let $T\colon \ell_1\to V$ be a bounded linear operator. Is it true that $T$ is not weakly compact if and only if there is a ...
1
vote
1answer
559 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 ...
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. ...
39
votes
4answers
2k views
Double sum - Miklos Schweitzer 2010
There is a question in the Miklos Schweitzer contest last year that keeps bugging me. Here it is:
Is there any sequence $(a_n)$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 ...
5
votes
3answers
2k views
Differential equations and Fourier and Laplace transforms
Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
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 ...
9
votes
2answers
271 views
Gelfand-Naimark Theorem
The Gelfand–Naimark Theorem states that an arbitrary C*-algebra $ A $ is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. There is another version, which states that ...
6
votes
1answer
187 views
Two different definitions of ellipticity
This is a question originating in another mathematics forum, matematicamente.it (in Italian).
In literature one encounters the word elliptic in (at least) two different definitions. In what follows ...
12
votes
1answer
623 views
Spectrum of a linear operator
Let $\ell^2 =\ell^2(\mathbb{Z})$. Choose $\theta \in ]0,1[$ and set:
$$Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{z}}$$
for each $x=(x_n)_{n\in \mathbb{Z}}\in \ell^2$ (thus $T$ is a convex ...
8
votes
1answer
164 views
Selfadjoint compact operator with finite trace
I have a compact selfadjoint operator $T$ on a separable Hilbert space. For some fixed orthonormal basis, the operator's diagonal is in $\ell^1(\mathbb{N})$.
Can we conclude that $T$ is trace ...
8
votes
1answer
252 views
A convergence problem in Banach spaces related to ergodic theory
Suppose $X$ is a Banach space, $T\in B(X)$, satisfied the following condition.
$\sup \Big\lVert\frac{1}{n}\sum \limits_{i=0}^{n-1}T^{i}\Big\rVert<\infty$
$\frac{1}{n}\lVert ...
6
votes
2answers
106 views
Question about Angle-Preserving Operators
This an exercise out of Spivak's "Calculus on Manifolds".
Edit: There was a typo in the exercise as is noted below in the answers. The statement has been edited to reflect this.
Given ...
5
votes
1answer
808 views
What is operator calculus?
I watched the excellent interview with Richard Feynman: http://www.youtube.com/watch?v=PsgBtOVzHKI In the interview Feynman mention that he at young age re-invented operator calculus.
I have searched ...
3
votes
1answer
230 views
Sets $f_n\in A_f$ where $f_{n+1}=f_n \circ S \circ f^{\circ (-1)}_n$ and operator $\alpha(f_n)=f_{n+1}$
Let's start with a function on the Reals (in this case for $x=0$ is not defined): for example $f(x)=b/x$, $x \in \mathbb R$
I define:
$$f_0:=f$$
$$f_{n+1}:=f_n \circ S \circ f^{\circ ...
2
votes
3answers
99 views
Norm of bounded operator on a complex Hilbert space.
It is fairly easy to show that for a bounded linear operator $T$ on a Hilbert space $H$ $$||T||=\sup_{||x||=1,||y||=1}|\langle y, Tx \rangle |.$$
If $H$ is a complex Hilbert space, can you show that
...
6
votes
1answer
89 views
A Marcinkiewicz approach
The problem was to prove the following that the operator
$$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 ...
6
votes
2answers
236 views
Unitisation of $C^{*}$-algebras via double centralizers
In most of the books I read about $C^{*}$-algebras, the author usually embeds the algebra, say, $A$, as an ideal of $B(A)$, the algebra of bounded linear operators on $A$, by identifying $a$ and ...
4
votes
1answer
121 views
Norms involving positive operators
Let's say we have $A \leq B$. Is it then true that $||Ax|| \leq ||Bx||$ (where $x, A, B$ all belong to the same finite-dimensional Hilbert space $H$)?
4
votes
1answer
160 views
Spectrum of this Operator
Let $A\colon \ell^{1}\to \ell^{1}$ be defined by $A(x)=(x_{2}+x_{3}+x_{4}+ \dots,x_1,x_2,x_3,\dots)$ where $x\in\ell^1$ iff $\sum|x_k|<\infty$.
Let $D$ be the closed unit disc in $\Bbb C$ and ...
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 $f_y(x)=B(x, y) \forall x \in E$. ...
4
votes
1answer
273 views
What is a Form Domain of an Operator?
I tried to look this up on Wikipedia, but I couldn't find anything.
I am reading Barry Simon's book "Schrödinger Operators", where he brings up the concept of a form domain $Q(A)$ of a ...
4
votes
0answers
296 views
Double dual of the space $C[0,1]$
The second dual or double dual of the space of all continuous functions on $[0,1]$, $C[0,1]$ is von Neumann algebra. Can anyone help me identifying this space?
4
votes
2answers
285 views
If $(I-T)^{-1}$ exists, can it always be written in a series representation?
If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm < $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$.
Thinking in terms of a ...
3
votes
2answers
203 views
Show that a finite-dimensional Banach space has a bijective compact operator
It is clear that if $ T: X \rightarrow X $ is a bijective compact operator, where $ X $ is a Banach space, then $ \dim(\text{Range}(T)) = \dim(X) $, which implies that $ \dim(X) $ must be $ < ...
3
votes
1answer
52 views
Pseudo Monotone Operator
Suppose $X$ is a real Reflexive Banach space. Let $A:X\rightarrow X^{\star}$ be a Pseudo Monotone operator, i.e. if $u_{n}\rightharpoonup u$ and $\limsup\langle Au_{n},u_{n}-u\rangle\leq 0$, then ...
3
votes
1answer
362 views
positive invertible operators
I need the following result. I think it's quite obvious but I don't know how to prove that: Let $C, T : \mathcal{H} \rightarrow \mathcal{H}$ be two positive, bounded, self-adjoint, invertible ...
